Succulent lobster replaces the lettuce in this luxurious take on the classic BLT. Hard-boiled eggs and avocado add heft to the sandwich (and welcome pops of color to the serving platter!).
In a large, deep pot lined with a steamer basket, bring 1 inch of water to a boil. Drop lobsters headfirst into pot; cover, reduce heat to medium, and steam until shells are bright red, 12 to 16 minutes, depending on size and shell thickness. Transfer to a tray; let cool slightly. Remove meat and cut into 1/2-inch slices.
Preheat oven to 400 degrees. Place bread on baking sheets and bake, flipping once, until lightly toasted, 10 to 12 minutes. Arrange tomatoes, lobster meat, eggs, avocados, basil, and toast on a large platter. Drizzle tomatoes with mignonette. Stir together butter and lemon juice; season with kosher salt and pepper and drizzle over lobster. For each sandwich, spread mayonnaise on toast, then top with tomato, lobster, avocado, egg, and basil. Sprinkle with flaky salt. Serve with more mignonette over top, if desired.
New-shell lobsters have a sweeter taste and more tender texture than thicker shelled ones. For instructions on removing the meat from the shells, watch this video.