In a small bowl, mix together mayonnaise, Old Bay seasoning, lemon juice, minced garlic, and Dijon mustard. Set aside to let the flavors meld.
Spread a generous layer of Old Bay aioli on the inside of each croissant half. Add the chopped lobster meat on top of the cheese. Add handful off cheddar cheese on top. Top with the other half of the croissant.
Preheat a skillet over medium-low heat. Spread butter on the outside of each croissant. Place the sandwiches on the skillet and press down with a spatula or another heavy pan. Cook for about 3-4 minutes per side, until the croissant is golden brown and crispy, and the cheese is melted. Serve and enjoy!
