Begin by tying the filet mignon pieces: Lay the meat on a cutting board and wrap a piece of twine around the outside of each piece. Pull the twine tightly so that the meat is secure and tie a knot. Sprinkle both sides of the meat generously with salt and pepper.
In a medium saute pan over high heat, add 2 tablespoons of the butter, the olive oil and garlic. When the butter begins to bubble, add the filet mignon pieces. Tip the pan slightly and baste the tops with the butter and olive oil. Cook until deeply browned, about 3 minutes, and then flip and continue cooking and basting on the other side. Next, turn the pieces on their sides and cook the edges until brown.
Add the leeks and 2 tablespoons of the butter and saute 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, lower the heat to simmer and add the truffle oil.
Place another saute pan on high heat. Spread the remaining 2 teaspoons butter onto the brioche halves and place in the pan until slightly browned, about 2 minutes.
To serve, slice the filet mignon on the bias and place some slices onto one half of each brioche bun. Top with some sauce, leeks and the other half of each bun. Serve immediately.
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