Sprinkle both sides of the beef with salt and pepper. Heat the olive oil in an oven-safe medium skillet over high heat. When the oil is hot, carefully place the filet in the pan. Cook until browned on both sides, about 3 to 4 minutes a side. Transfer the steak to the oven and bake until a meat thermometer reads 130 degrees F. for medium-rare, about 5 to 6 minutes. Remove the beef from the oven and let rest for 5 minutes.
For the sauce:
Heat the olive oil in a medium skillet over medium heat. Add the mushrooms and the shallots and cook until golden brown and tender, about 5 minutes. Add the thyme, white wine, salt, and pepper and continue to cook until all of the liquid has evaporated, about 5 more minutes.
Meanwhile, put the Gorgonzola, mayonnaise, and mustard in a food processor and combine until smooth. Transfer the Gorgonzola mixture to the skillet with mushrooms and shallots. Gently stir the cheese mixture into the mushroom mixture.