Make aioli: In a small bowl, mix together mayonnaise, garlic powder, lemon juice and chives.
Set aside to let the flavors come together.
Make the brown butter lobster: Melt butter on low heat, stirring often. After a few minutes the butter will start to foam and smell like caramelization. Once it starts to brown, add lobtster and season with salt and pepper.
Cook lobster in 2 tbsp of the butter on low for about 4-5 minutes, just to get it nice, warmed and combined.
The rest: Add aioli onto the brioche along with buttery lobster. Grate cheese and add overtop. Close sandwich and add back into the skillet with the last tbsp of butter. Cook on low with a top, flipping once the bread is nice and browned. Cook until cheese is melted. Slice and enjoy!