Remove the filet mignon from the refrigerator and let it come to room temperature for about 30 minutes. Preheat your oven to 400°F (200°C).

In a small bowl, combine the crushed pink peppercorns, chopped thyme, salt, and black pepper. Pat the filet mignon dry with paper towels and rub the peppercorn mixture all over the steaks, pressing gently to adhere.

In an oven-safe skillet, heat the olive oil over medium-high heat. Once the oil is hot, add the filet mignon and sear for about 3-4 minutes on each side, or until a nice crust forms.

Transfer the skillet to the preheated oven and roast for an additional 5-7 minutes for medium-rare, or until your desired doneness is reached. Use a meat thermometer to check the internal temperature (130°F for medium-rare).

While the steaks are roasting, melt the butter in a separate saucepan over medium heat. Add the chopped shallots and cook until they are translucent and fragrant, about 3-4 minutes.

Stir in the beef stock and balsamic vinegar, bringing the mixture to a simmer. Cook for about 5-7 minutes, allowing the sauce to reduce and thicken slightly.

Once the steaks are done, remove them from the oven and let them rest for 5 minutes. Slice the filet mignon and drizzle the brown butter shallot reduction over the top before serving.