Heat a heavy-bottomed pan over medium-high to high heat. Add the butter and allow to melt. Generously coat each side with kosher salt and pepper. Add filets to the hot pan. Cook 3-4 minutes, allowing for the cooking side to create a nice browned crust, and then flip. Cook for an additional 3-4 minutes on this side. Set aside on a clean plate and cover with aluminum foil while you cook your sauce.
Heat a clean sauce pan over medium heat. Add the butter and allow to melt. Add shallot and cook for 2 minutes, stirring occasionally. Add garlic and cook for one minute. Add gorgonzola and stir to help melt. Once cheese as melted, add heavy cream, 2 tbsp. at first, and stir to incorporate. If you'd like a thinner sauce, add the additional tbsp. of cream. Season with kosher salt and ground black pepper to taste. Drizzle over filet mignon and serve.