When Judge Aileen M. Cannon presides over a hearing on Friday in former President Donald J. Trump’s classified documents case, she will spend the day considering well-trod arguments about an arcane legal issue in an unorthodox manner.

It will be the latest example of how her unusual handling of the case has now become business as usual.

Over the past several months, Judge Cannon, who was appointed by Mr. Trump in his final days in office, has made a number of decisions that have prompted second-guessing and criticism among legal scholars following the case. Many of her rulings, on a wide array of topics, have been confounding to them, often evincing her willingness to grant a serious hearing to far-fetched issues that Mr. Trump’s lawyers have raised in his defense.

The issue that will be discussed on Friday in Federal District Court in Fort Pierce, Fla., is a motion by the defense to dismiss the charges in the case on the grounds that Jack Smith, the special counsel who filed them last spring, was improperly funded and appointed.

The defense has argued that Mr. Smith was not named to his post by the president or approved by the Senate like other federal officers, and that Attorney General Merrick B. Garland, who gave him the job, had no legal power to do so on his own.

Mr. Smith’s deputies have countered that under the appointments clause of the Constitution, agency heads like Mr. Garland are authorized to name “inferior officers” like special counsels to act as their subordinates.

And while the subject of the hearing may seem rather technical, what is most unusual is that it is happening at all.

Reaching back to the early 1970s, courts have repeatedly rejected efforts like Mr. Trump’s to question the legality of independent prosecutors. Those have included the Supreme Court upholding the appointment of Leon Jaworski, one of the special prosecutors who investigated the Watergate scandal, in a decision that was largely focused on the issue of President Richard Nixon’s claims of executive privilege.

Judges have also tossed efforts to invalidate the work of special counsels like Robert S. Mueller III, who examined connections between Russia and Mr. Trump’s 2016 campaign, and David C. Weiss, who has brought two criminal cases against Hunter Biden, President Biden’s son.

Despite this record, however, Judge Cannon has decided to consider the constitutionality of Mr. Smith’s appointment anew — and not on the merits of written briefs, but rather at an expansive hearing that will spill across two days. The proceeding might go beyond the normal process of merely making arguments and could include, as the judge recently wrote, the “presentation of evidence,” though it remains unclear what evidence she meant.

In another unusual move, Judge Cannon is allowing three lawyers who have filed what are known as amicus or friend-of-the-court briefs to argue in front of her for 30 minutes each. While these outside parties — referred to as “amici” — are commonly permitted to make their case directly to judges in appellate courts like the Supreme Court, that is not the standard practice in trial courts.

“The fact that Judge Cannon granted the amici request for oral argument seems to suggest that she is seriously considering the constitutional argument against the appointment of the special counsel,” said Joel S. Johnson, an associate professor at Pepperdine Caruso School of Law.

One of the most striking aspects of Judge Cannon’s tenure is that she has largely ignored a common practice in the Southern District of Florida, where she sits, of trial judges handing off routine motions to the magistrate judge attached to a case.

Judge Cannon has not delegated any motions to the magistrate judge in this case, Bruce E. Reinhart. And Judge Reinhart knows the case well, having approved the search warrant used by the F.B.I. two years ago when agents descended on Mar-a-Lago, Mr. Trump’s estate in Palm Beach, and hauled away a trove of classified material that is central to the case.

Even before Mr. Trump was indicted last June on charges of illegally holding on to classified documents after he left office and then obstructing the government’s repeated efforts to retrieve them, Judge Cannon made an unusual move to insert herself into the case unnecessarily.

After the F.B.I. searched Mar-a-Lago, she barred federal prosecutors from using any evidence collected from Mr. Trump’s estate until an independent arbiter sorted through it for materials that were privileged. That decision was quickly reversed with a stern rebuke from the appeals court that sits over her.

In recent months, Judge Cannon has continued in much the same vein, making several quizzical decisions or just as often putting off making them at all.

In February, for example, she stunned observers of the case when she agreed to let Mr. Trump’s lawyers reveal the names of several government witnesses in a filing they were planning to submit. Mr. Smith’s deputies, fearful for the safety of the witnesses, asked her to revisit her decision, which she ultimately did.

One month later, the judge issued another baffling order, asking the defense and prosecution to send her draft instructions for the jury that seemed to adopt one of Mr. Trump’s key defenses in the case.

The order was bizarre on its face because questions surrounding jury instructions are usually hashed out on the eve of trial and at that point Judge Cannon had not set a trial date yet.

It was even stranger because by appearing to adopt Mr. Trump’s defense, the judge seemed to be nudging any eventual jurors toward acquitting the former president or leaving open the possibility that she herself could acquit him near the end of the proceeding by declaring that the government had failed to prove its case.

More recently, Judge Cannon held a hearing to consider giving Mr. Trump’s two co-defendants, Walt Nauta and Carlos De Oliveira, what is known as a bill of particulars, a detailed recitation of the charges supplementing those laid out in the indictment.

Such documents are almost never granted to criminal defendants. And while Judge Cannon ultimately denied the requests, her decision to open her courtroom for a hearing on the issue in the first place was unusual, as other district judges (or magistrate judges) often decide such minor pretrial issues on the basis of court filings alone. In this case, however, Judge Cannon has made holding hearings the norm.

Last month, Judge Cannon issued an order formally scrapping the May 20 trial date in the case. While judges enjoy wide latitude over the timing of cases and do not generally need to justify their scheduling decisions, Judge Cannon cited an odd authority for putting off the trial: a 2013 essay titled “The National Security Trials: A Judge’s Perspective.”

The author was T.S. Ellis III, a respected former jurist who sat for years in the Eastern District of Virginia. But it was nonetheless unusual for a judge to cite an informal piece of writing as a basis for a decision rather than relying solely on legal cases.

“It’s an interesting sign about the work that’s going into writing these decisions,” said Tracey E. George, a professor at the Vanderbilt Law School. “Or the lack of work, the lack of care.”