Remove the filet mignon steaks from the refrigerator and let them sit at room temperature for 15 minutes before cooking.
Season both sides of the steaks evenly with kosher salt and freshly ground black pepper.
Heat olive oil and 1 tablespoon of butter in a large skillet over medium-high heat until the butter is melted and the pan is hot but not smoking.
Add the filet mignon steaks to the skillet and sear for 4 minutes on the first side without moving them.
Flip the steaks and cook for an additional 4 minutes for medium-rare doneness, or longer to your preferred doneness.
Remove the steaks from the skillet and transfer them 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 sauté for 5 minutes until they release their moisture and begin to brown.
Add the finely chopped shallots and minced garlic to the mushrooms and cook for 2 minutes until fragrant and softened.
Pour in the dry white wine and scrape the bottom of the skillet with a wooden spoon to deglaze, cooking until the wine is reduced by half, about 3 minutes.
Add the beef broth and continue to cook for 3 minutes to reduce slightly.
Stir in the heavy cream and fresh thyme leaves, and simmer the sauce for 2 minutes until it thickens slightly.
Return the filet mignon 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.