How to Make It
Preheat the oven to 450°. Arrange the bacon slices on 2 large rimmed baking sheets and bake for about 15 minutes, or until browned and chewy. Drain on paper towels.
In a medium bowl, whisk the egg yolks. Very gradually whisk in 1 cup of the olive oil; start with small drizzles, then whisk in the oil in a thin stream until the mayonnaise is very thick. Whisk in the lemon juice and garlic paste and season with salt, pepper and cayenne.
Spread the tomato slices on a large plate. Season with salt and pepper and sprinkle with the basil and the remaining 2 tablespoons of olive oil.
Heat a large cast-iron skillet. Spread each slice of brioche with 1 teaspoon of butter and set 2 slices in the skillet, buttered side down. Toast over high heat until browned and crisp. Turn and toast the other side. Repeat with the remaining slices of brioche.
Set the brioche toasts on a work surface, buttered side up. Spread each toast with 1 tablespoon of the mayonnaise. Top 6 of the toast slices with 2 lettuce leaves, 2 tomato slices and 4 bacon strips and close the sandwiches. Carefully cut the BLTs in half and serve.