1. Heat butter with garlic in a small saucepan over medium heat until melted. Simmer 1 minute. Stir in Old Bay and season with salt. Squeeze juice of halved lemon into butter mixture (reserve halves). Remove from heat; cover to keep warm.

2. Cover potatoes with 2 inches cold water in a large pot; bring to a boil. Add 2 tablespoons salt, reduce heat to a simmer, and cook until potatoes are just tender when pierced with the tip of a sharp knife, about 12 minutes. Add corn and squeezed lemon halves; return to a boil, then drain.

3. Preheat grill to medium-high. Brush grates with oil. Toss potatoes, corn, shrimp, and sliced lemons with 2 tablespoons butter mixture in a large bowl. Grill, turning occasionally, until shrimp are cooked through and potatoes, lemons, and corn are marked in spots, about 3 minutes for shrimp, 4 to 5 minutes for vegetables and lemons. Transfer to a large bowl; toss with 1/2 cup butter mixture. Serve with remaining butter mixture on the side.