Heat the oven to broil and arrange a rack in the upper third.

Heat the oil in a large frying pan over medium heat until shimmering. Add the onions and garlic and cook, stirring occasionally, until the onions have softened, about 6 to 8 minutes. Remove the pan from the heat and let cool slightly.

Tear the bread into rough 1-inch pieces and place in a large bowl. Add the milk and stir until the mixture forms a wet mash. Let sit until the bread absorbs all of the milk, about 5 minutes. Add the slightly cooled onions and garlic, as well as the elk, pork, salt, pepper, and 3/4 teaspoon of the allspice. Using your hands, mix until just combined.

Fill a medium bowl with water. Set aside two rimmed baking sheets. Form the meat mixture into about 48 (1-1/4-inch) meatballs, wetting your hands as necessary to keep the mixture from sticking, and place them about 1/2 inch apart on the baking sheets.

Place the tomato purée, vinegar, bourbon, brown sugar, ketchup, molasses, Worcestershire, chili powder, and remaining 3/4 teaspoon allspice in a large pot or Dutch oven and whisk to combine. Bring the mixture to a boil over medium-high heat. Reduce the heat to low and simmer.

Meanwhile, place 1 sheet of meatballs in the oven and broil until browned all over, about 10 minutes. Transfer the meatballs to the pot of simmering sauce and gently stir to coat. Repeat with the second sheet of meatballs.

Simmer, carefully stirring the meatballs occasionally, until the sauce has thickened slightly and the meatballs are fully cooked, about 30 to 40 minutes.