In a small saucepan over medium heat, melt the 1/2 cup unsalted butter. Add the smashed garlic, 1 tsp sea salt, and 1/4 tsp black pepper and simmer about 1 minute until fragrant.
Stir in the chopped dill and 2 Tbsp lime juice, then immediately remove the pan from the heat to prevent the garlic from burning.
Place the salmon fillets skin-side down on a platter and spoon about one-third of the butter mixture over the tops; use a brush or spoon to coat evenly. Let the salmon sit while you preheat the grill.
Preheat the grill to medium (about 350–400°F) and oil the grates to prevent sticking.
Place the salmon on the preheated grill skin-side up. Cover and cook undisturbed for about 3 minutes on the first side.
Carefully flip the fillets using a spatula and tongs, brush with another one-third of the butter mixture, then cover and cook about 2 more minutes, or until the internal temperature reaches about 140°F (it will rise to 145°F off the grill).
Transfer the salmon to a serving platter, drizzle with the remaining butter sauce (reheat briefly if it has solidified), garnish with additional dill, and serve with lime wedges.