Pour water into a large pot to a depth of 1 inch; bring to a boil and salt generously. Add lobsters, cover, and cook until bright red, 8–10 minutes. Transfer lobsters to a rimmed baking sheet and let cool.
Crack lobster shells, pick meat from tail and claws, and cut into 1/2-inch pieces. Mix lobster, celery, lemon juice, chives, and 2 tablespoons mayonnaise in a medium bowl; season with salt and pepper and add more mayonnaise, if desired.
Heat a large skillet over medium heat. Spread flat sides of buns with butter. Cook until golden, about 2 minutes per side; fill with lobster mixture.
DO AHEAD: Lobster meat can be prepared 1 day ahead. Cover and chill. Toss with remaining ingredients just before serving.