Prepare burgers as recipe directs. Keep patties refrigerated until ready to cook.
Warm oil in a large skillet over medium-high heat. Add onion and cook, stirring often, until softened, 3 to 5 minutes. Add mushrooms, season with salt and pepper and sauté until mushrooms release their liquid and it evaporates, about 12 minutes. Stir in parsley, lemon juice and thyme. Season with additional salt and pepper, if desired.
Grill or broil burgers as recipe directs. Top with cheese during last minute of cooking time. Transfer to buns, and spoon mushroom mixture over each. Serve immediately with assorted toppings, if desired.