1. Preheat oven to 350 degrees. In a large saucepan, melt butter over medium-low heat; do not let the butter start to bubble. Add lobster tails to butter and simmer until cooked through, about 5 minutes. Remove tails from butter, reserving butter. Let tails cool slightly, then pick meat from shells, reserving shells, and cut meat into 4 to 5 pieces per tail.
2. In a medium saucepan, combine vegetable stock with reserved tail shells. Bring to a simmer and cook over medium heat for 30 minutes. Strain shells and discard. Keep stock warm.
3. Pour 2½ tablespoons of lobster-poaching butter into a skillet over medium heat. (Save remaining butter for another use.) Add fennel, shallot, and garlic, and cook until translucent and fragrant, being careful not to let it brown. Season with salt. Remove from heat and allow to cool to room temperature.
4. In a large bowl, combine mayonnaise, dill, lemon zest, and vegetable mixture. Mix until fully incorporated. Add brioche, lobster meat, and strained vegetable stock; stir gently to combine until brioche pieces are evenly coated. Transfer mixture to a large casserole dish. Bake for 1 hour, or until dressing is set and bubbling with very little moisture around the edges.