In a small skillet over medium heat, add 1 tablespoon of olive oil and sauté the chopped shallots until they are soft and caramelized, about 8-10 minutes. Stir occasionally to prevent burning.

Once caramelized, remove from heat and let cool slightly. In a bowl, combine softened butter, caramelized shallots, chopped rosemary, and lemon juice. Mix until well combined. Season with salt and pepper to taste.

Transfer the compound butter onto a piece of plastic wrap, shape it into a log, and refrigerate until firm.

Preheat your oven to 400°F (200°C). Season the filet mignon steaks generously with salt and black pepper on both sides.

In a large oven-safe skillet, heat the remaining 1 tablespoon of olive oil over medium-high heat. Add the minced garlic and fresh thyme, stirring quickly to prevent burning.

Add the seasoned filet mignon steaks to the skillet. Sear each side for 3-4 minutes until a golden-brown crust forms.

Transfer the skillet to the preheated oven and roast for about 6-8 minutes for medium-rare, or until the desired doneness is reached. Use a meat thermometer for accuracy (130°F for medium-rare).

Remove the skillet from the oven and let the steaks rest for 5 minutes.

Slice the chilled compound butter into rounds and place a slice on each filet mignon. Allow the butter to melt over the warm steaks before serving.

Garnish with additional thyme if desired and serve with your favorite sides.