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 olive oil and 1 tablespoon of butter in a large skillet over medium-high heat until hot but not smoking.
Add the filet mignon 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 transfer to a plate; cover loosely with foil to keep warm.
Reduce the heat to medium and add the remaining 1 tablespoon of butter to the skillet.
Add the sliced mushrooms and cook, stirring occasionally, until they release their moisture and begin to brown, about 5 minutes.
Add the chopped shallot and minced garlic to the mushrooms and cook for 1-2 minutes until fragrant.
Pour in the white wine and scrape the bottom of the skillet to deglaze, cooking until the wine is reduced by half, about 3 minutes.
Add the beef broth and continue to cook until the liquid is reduced by half, about 5 minutes.
Stir in the heavy cream and fresh thyme leaves, and simmer the sauce until it thickens slightly, about 3 minutes.
Return the steaks to the skillet and spoon the mushroom sauce over them to warm through for 1-2 minutes.
Serve the filet mignon steaks topped with the mushroom sauce immediately.