For the beef tenderloin: preheat the oven to 350°f. Season tenderloin with ½ teaspoon salt and ½ teaspoon pepper. Heat ¼ cup olive oil in a large, heavy braising pan over high heat.
When the oil is hot, carefully place the tenderloin in the pan. Cook until browned on all sides, about 3 to 4 minutes a side.
Transfer to a foil-lined baking sheet and bake until a meat thermometer reads 140°f for medium-rare, about 20 minutes. Remove the beef from the oven, loosely cover with foil, and let the beef rest for 10 to 15 minutes.
For the sauce: heat the remaining 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, ½ teaspoon salt, and ½ teaspoon pepper and continue to cook until all of the liquid has evaporated, about 5 more minutes. Let cool slightly.
Put the Gorgonzola cheese, mayonnaise, and mustard in a food processor and combine until smooth. Transfer the Gorgonzola cheese mixture to a medium bowl.
Gently, stir the mushroom mixture into the cheese mixture. Slice the beef tenderloin into 1-inch thick slices and serve on a platter.
Transfer the cheese sauce to a serving bowl and invite guests to dollop their beef with the cheese sauce.