SEASON THE FILETS generously with salt and pepper. In a medium cast-iron skillet over medium-high heat, melt 2 tablespoons of the butter with the olive oil and garlic. When the butter begins to bubble, add the filet mignon pieces. Tip the skillet slightly and baste the tops with the butter and olive oil. Cook until deeply browned on the bottom, about 3 minutes. Flip and continue cooking and basting on the other side until brown, about 3 minutes. Turn the pieces on their sides and sear the edges until brown. Remove from the skillet and let the steak rest while you prepare the leeks and sauce.
Add the leeks and 2 tablespoons of the butter to the skillet and sauté until the leeks are soft, about 4 minutes. Add the tarragon. Turn the heat to high and add the red wine. Let the sauce reduce by half, 2 to 3 minutes. Lower the heat to simmer and stir in the truffle oil.
Heat a large skillet over medium-high heat. Spread the remaining 2 teaspoons butter onto the brioche buns and place in the pan until slightly browned, about 2 minutes. To serve, slice the filet mignon against the grain and place on one half of each brioche bun. Top with some leeks and their sauce and the other half of each bun and serve immediately.