Eleven days into deliberations, the jury has told Judge Sarmina that they're hung on all but one of the counts. Now what?
That doesn't mean it's over, because the Court can still give what's informally known as a "dynamite" charge and try to move the jury to a verdict one way or the other. Under the century-old United States Supreme Court case, Allen v. United States, 164 U.S. 492 (1896), the Court may admonish the jury to keep trying, and can ask jurors to reconsider their positions. In Pennsylvania, the charge is known as a Spencer charge after the main case applying it here, Commonwealth v. Spencer, 442 Pa. 328, 275 A.2d 299 (1971), which incorporated American Bar Association standard 15-5.4. That standard says:
(a) Before the jury retires for deliberation, the court may give an instruction which informs the jury:
(1) that in order to return a verdict, each juror must agree thereto;
(2) that jurors have a duty to consult with one another and to deliberate with a view to reaching an agreement, if it can be done without violence to individual judgment;
(3) that each juror must decide the case for himself or herself but only after an impartial consideration of the evidence with the other jurors;
(4) that in the course of deliberations, a juror should not hesitate to reexamine his or her own views and change an opinion if the juror is convinced it is erroneous; and
(5) that no juror should surrender his or her honest belief as to the weight or effect of the evidence solely because of the opinion of the other jurors, or for the mere purpose of returning a verdict.
(b) If it appears to the court that the jury has been unable to agree, the court may require the jury to continue their deliberations and may give or repeat an instruction as provided in section (a). The court should not require or threaten to require the jury to deliberate for an unreasonable length of time or for unreasonable intervals.
(c) The jury may be discharged without having agreed upon a verdict if it appears that there is no reasonable probability of agreement.The core concern in giving a Spencer instruction is to ensure the jury doesn't feel coerced by the court to reaching a verdict one way or another. The charge cannot, for example, "instruct the minority jurors to yield to the majority," nor instruct that "the majority  need not re-examine their position." Commonwealth v. Schaffer, 2005 PA Super 14 (Pa. Super. Ct. 2005). The point of the charge is to make sure everyone has thoroughly considered the issues and so has reached a fully informed and honest belief as to the evidence.
We can expect Judge Sarmina to grant a Spencer charge here, perhaps after trying other options like further explaining certain legal issues, if she believes the jury might be stuck on their understanding of the jury instructions. By this point, though, the jury has probably heard enough about the law of child endangerment; they're likely stuck on factual interpretations, most likely Monsignor Lynn's state of mind. The defense lawyers will inevitably object to the Spencer charge, as would be proper to preserve their appeal options, but the charge will likely be given nonetheless as a last resort.
Importantly, a hung jury does not necessarily benefit Monsignor Lynn. As the United States Supreme Court has repeatedly affirmed, most recently in Yeager v. United States, 129 S. Ct. 2360 (2009), the "double jeopardy" clause of the Constitution does not bar re-trial of a defendant where the prior trial ended in a hung jury:
[W]hile the defendant has an interest in avoiding multiple trials, the [Double Jeopardy] Clause does not prevent the Government from seeking to reprosecute. Despite the argument's textual appeal, we have held that the second trial does not place the defendant in jeopardy "twice." Richardson v. United States, 468 U.S. 317, 323, 104 S.Ct. 3081, 82 L.Ed.2d 242 (1984); see 3 J. Story, Commentaries on the Constitution § 1781, pp. 659-660 (1833). Instead, a jury's inability to reach a decision is the kind of "manifest necessity" that permits the declaration of a mistrial and the continuation of the initial jeopardy that commenced when the jury was first impaneled. See Arizona v. Washington, 434 U.S. 497, 505-506, 98 S.Ct. 824, 54 L.Ed.2d 717 (1978); United States v. Perez, 9 Wheat. 579, 580, 6 L.Ed. 165 (1824). The "interest in giving the prosecution one complete opportunity to convict those who have violated its laws" justifies treating the jury's inability to reach a verdict as a nonevent that does not bar retrial. Washington, 434 U.S. at 509, 98 S.Ct. 824.Id. If it's a hung jury, the only question is if the Philadelphia District Attorney declines to re-try Monsignor Lynn. That's possible, but in my opinion unlikely.