In a large bowl, combine the turkey, steak sauce, thyme, salt and pepper. Shape the mixture into 5 patties. Cook the patties, in batches if necessary, in a skillet over medium heat until cooked through, about 6 minutes per side.
In a large skillet, melt the 2 tablespoons butter over medium heat. Add the mushrooms and onion and cook until the onion is tender, 8 to 10 minutes, stirring frequently. Remove the mixture from the skillet; set aside and keep warm.
Butter the bread on both sides and place on griddle and grill both sides until toasted. Remove and top with bacon jam if desired.
Place the toasted bread in a single layer on a baking sheet. Top each with 1 patty. Divide the onion mixture among the patties, and top each with 1 slice cheese. Broil until the cheese melts.