Remove the filet mignon steaks from the refrigerator and let them sit at room temperature for 15 minutes.
Season both sides of the steaks evenly with salt and freshly ground black pepper.
Heat 1 tablespoon of butter and the olive oil in a large skillet over medium-high heat until hot but not smoking.
Add the steaks to the skillet and sear for 4 minutes on each side for medium-rare, or until desired doneness is reached. Remove the steaks from the skillet and set aside to rest, tented loosely with foil.
Reduce the heat to medium and add the remaining 1 tablespoon of butter to the skillet.
Add the chopped shallots and sauté for 2 minutes until softened.
Add the sliced mushrooms and cook for 5 minutes until they release their moisture and become golden brown.
Sprinkle the flour over the mushrooms and stir well to combine. Cook for 1 minute to remove the raw flour taste.
Pour in the white wine and beef broth, scraping the bottom of the skillet to release any browned bits.
Simmer the sauce for 5 minutes until it thickens slightly.
Stir in the heavy cream and fresh thyme leaves, and cook for an additional 2 minutes until the sauce is creamy and heated through.
Return the steaks to the skillet and spoon the mushroom sauce over them to warm through for 1-2 minutes.
Serve the filet mignon topped with the mushroom sauce immediately.