Preheat a grill to cook over high heat.
Pat the fish dry with paper towels and place it vertically on a clean cutting board with the head pointing towards you. Using a very sharp, thin knife, make an incision just behind and above the anal and dorsal fins, parallel to the spine. Using the top of the ribs as a guide and leaving the fillet attached by about an inch of flesh along the top of the fish, cut all the way to the head. When you reach the head, gently but firmly smack the dull side of the knife to cut through the skull, pulling the knife out between the eyes and the gills. (The fish will be effectively “half” butterflied at this stage, with one fillet detached from the bones but still connected along the top of the fish.) Gently wash the fish with cold water to remove any remaining viscera and again pat dry with paper towels. Cut 4 shallow incisions along the rib bones without piercing the skin and insert the garlic cloves; season with salt and pepper. Halve one lemon and squeeze the juice of one half over the fish; cut the other half into thin rings and distribute over the fish.
Place a large cast iron skillet on the grill to preheat. To a medium bowl, add the Brussel sprouts and 2 tablespoons olive oil, season to taste with salt, and toss to combine. When the skillet is hot, add half of the Brussels sprouts and cook, stirring frequently until lightly browned all over, about 10 minutes. Return the cooked Brussels sprouts to the bowl and toss with the uncooked ones, then sprinkle evenly over the interior of the fish. Drizzle with the remaining olive oil and fold the top fillet back down to enclose.
Place the fish on the grill, thick-side down, and cook without turning until the skin is browned and the meat can be pierced without any resistance, around 30 minutes. Using a wide, metal spatula to turn, and using your hand to help keep the fish closed, carefully flip the fish and continue cooking until the meat is opaque and yielding, about 15 minutes more. Carefully transfer to a tray and serve hot.