Use this to make our Stuffed Beef Burgers.
Melt butter with oil in a medium skillet over medium-high heat. Add mushrooms, salt, and pepper. Raise heat to high. Cook, stirring, until mushrooms are browned and softened, about 5 minutes. Stir in thyme; cook 30 seconds.
Transfer to a bowl; let cool completely. Stir in cheese.
Filling can be refrigerated in an airtight container, up to 1 day.