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.