Heat an outdoor grill for medium, direct heat. Meanwhile, cut 1 medium lemon into 4 wedges. Cut 2 tablespoons unsalted butter into 4 pieces.
Pat 4 red snapper fillets dry with paper towels. Rub with 1 tablespoon olive oil and season all over with 1 1/2 tablespoons Cajun or Old Bay seasoning. Line a rimmed baking sheet with aluminum foil. Bring the baking sheet, fish, butter, and flat spatula out to the grill.
Scrape the grill grates clean if needed. Oil the grill grates with a folded paper towel dipped in olive oil. Place the fish on the grill grates (skin side down if skin-on). Cover and grill until the bottoms turn opaque and a flat spatula can slide underneath the fillets with little resistance, about 2 minutes. Flip the fillets, cover, and grill until almost completely opaque and the center of each registers at least 125ºF, 2 to 3 minutes more.
Transfer the fillets to the baking sheet. Top each fillet with a piece of butter, then pull up the sides of the foil to meet in the middle and cover the red snapper completely. Let sit for 5 minutes: The residual heat will increase the internal temperature to about 130ºF and also create steam to keep the fish moist. Uncover and serve with the lemon wedges for squeezing and more Cajun or Old Bay seasoning as needed.
Recipe Note
Cajun Seasoning or Old Bay substitute: To make your own Cajun seasoning at home, combine 1 teaspoon kosher salt, 2 teaspoons paprika, 1 teaspoon dried thyme, 1/2 teaspoon garlic powder, 1/2 teaspoon onion powder, 1/2 teaspoon black pepper, and 1/4 teaspoon cayenne pepper (optional). This makes about 3 tablespoons seasoning.
