Set up a grill for cooking with direct and indirect heat. If using a charcoal grill, build the coals on one side only. If using a gas grill, heat one side only.
Make the aioli: In a mixing bowl whisk together the mayo, cilantro, garlic, lime zest and juice. Season with salt and pepper and set aside. (If serving the fish later in the day, let the aioli chill in the refrigerator until ready to serve.)
Pat the fish dry with paper towels, then sprinkle with salt. Brush a cast-iron pan with 1/2 tablespoon butter. Place the fish, skin-side down, in the pan and place the pan on the direct heat side of the grill. Cook until the skin is crisp and has released from the pan, 3 to 4 minutes. Flip the fish and add the remaining 2 1/2 tablespoons butter. Move the pan to the indirect heat side of the grill. Let the butter melt and lightly brown, then baste the fish for 1 to 2 minutes. Add the lemon juice and continue basting for another minute. Close the lid of the grill and cook until the fish is cooked through, about 3 more minutes. Remove from the grill and spoon the remaining brown butter in the pan over the fish. Serve with the aioli, cilantro and lime wedges.
