Place lobster, mayonnaise, olive oil, shallot, chives, lemon juice, parsley, salt, and pepper in a bowl; mix with a spatula until combined. Cover and place in the refrigerator.
Melt butter in a large skillet. Add garlic; cook and stir until fragrant, about 1 minute. Add hot dog buns; fry until golden brown, about 5 minutes.
Fill each roll with an equal amount of lobster mixture and 1/4 cup shredded lettuce.
