Sauté onion in 2 tablespoons butter in a large skillet over medium heat for 10 minutes, or until soft and lightly browned. Remove from heat; allow to cool 15 minutes.
Add sautéed onions along with the turkey, breadcrumbs, salt, thyme, pepper, egg, and 2 tablespoons of butter to the food processor fitted with a steel blade. Process 15 seconds; scrape down sides. Process an additional 15 seconds.
Divide mixture into 6 equal pieces (around 4.51 ounces each). (See Note 4). Shape into patties about 1/2-inch thick. (See Note 5)
Place remaining tablespoon of butter and olive oil in a large skillet set over medium heat. When the butter foam begins to subside, add the patties and cook for 5 minutes. Turn and cook for 5 minutes more (a total of 10 minutes). Place the patties on paper towels to drain, and keep warm while finishing the sauce.
Add chopped onion to same skillet used to cook the turkey burgers. Sauté over medium heat for 3 to 4 minutes without browning. Increase the heat to medium high; add mushrooms and sauté for a minute or two without browning.
Add the Vermouth and Madeira. Increase heat to high and boil down quickly until the liquid is syrupy, 2 to 3 minutes. Stir in the cream; boil down again over high heat until the cream has thickened slightly, 1 to 2 minutes.
Pour Mushroom Cream Sauce over turkey burgers and serve. (See Note 6) Yield 1 1/2 cups.