Prepare the Mushroom Sauce:
In a large skillet, heat 2 tbsp butter and 1 tbsp oil over medium-high heat. Add mushrooms and sauté for 6 minutes until softened and most of the moisture evaporates.
Add onion and cook for 3 more minutes, then stir in garlic, thyme, ¼ tsp salt, and ⅛ tsp pepper. Cook for 2 minutes until fragrant.
Transfer mixture to a plate. Wipe the pan clean.

Cook the Filet Mignon:
Season steaks with 1 tsp salt and ¼ tsp pepper.
In the same pan, melt 2 tbsp butter with 1 tbsp oil. Sear steaks for 3-5 minutes per side for medium-rare (5-6 minutes for medium). Remove and place with the mushrooms.

Make the Sauce:
Add beef broth and boil for 5-7 minutes until reduced to about ⅔ cup.
Stir in whipping cream and cook for 2 minutes until slightly thickened. Adjust seasoning.

Finish the Dish:
Return steaks and mushrooms to the pan, spoon sauce over, and heat for 1-2 minutes. Garnish with thyme and serve immediately.