Defrost the puff pastry sheet according to package instructions.
Preheat oven to 375°F.
Heat a skillet over low–medium heat. Add 2 tablespoons butter and the thinly sliced onion. Cook until caramelized, then stir in chopped parsley (if using) and remove from heat.
In a bowl, season the ground antelope with fennel, garlic powder, salt, and pepper. Shape into 4 equal-sized patties and set aside.
On a lightly floured surface, roll out the puff pastry sheet and cut into 4 equal squares.
In the center of each square, layer a slice of Canadian bacon, a slice of mozzarella, caramelized onions, and then an antelope patty.
Fold the corners of the pastry up and around the burger, then carefully flip so the seam is on the bottom. Place on a parchment-lined baking sheet.
Melt the remaining 2 tablespoons of butter and brush over the tops of the wrapped burgers. Sprinkle with sesame seeds, a pinch of salt, and a bit of black pepper.
Bake for 35-45 minutes, or until the puff pastry is golden brown and the filling is cooked through.
Let cool slightly before serving.