Prepare the following, adding each to the bowl of a food processor fitted with a blade attachment as it is completed: Finely grate 1 ounce Pecorino Romano cheese (about ½ packed cup, or use ⅓ cup store-bought grated). Juice ½ medium lemon (about 1 ½ tablespoons). Smash and peel 2 garlic cloves. you h Pick the leaves from 1 large bunch fresh basil until you have 2 lightly packed cups.

Add ⅓ cup pine nuts, 1 teaspoon kosher salt, and ½ teaspoon black pepper. Pour in ¼ cup olive oil and pulse until a smooth paste forms, about 12 (1-second) pulses. If dry, add more olive oil, 1 tablespoon at a time, and pulse again until the pesto is smooth.

Heat an outdoor grill for direct, medium heat (350 to 375ºF). Meanwhile, cut ½ medium lemon into 4 wedges and cut 2 tablespoons salted butter into 4 pieces.

Line a large plate with a large sheet of aluminum foil. Pat 4 halibut fillets dry with paper towels, then place on second unlined large plate. Coat the fish on both sides with 2 tablespoons olive oil and season with 1 teaspoon kosher salt and 1 teaspoon black pepper.

Scrape the grates clean if needed. Oil the grill grates with a paper towel dipped in olive oil. Place the fish on the grill grates (skin-side down if there is skin). Cover and grill until the bottoms turn opaque and a spatula can slide underneath the fillets with little resistance, about 3 minutes. Flip the fillets, cover, and grill until they are almost completely opaque and an instant-read thermometer inserted into the center of the thickest fillet registers 125ºF, 3 to 4 minutes.

Transfer the fillets to the foil-lined plate in a single layer. Quickly place 1 piece of butter onto each filet. Fold the foil up and over the halibut to completely cover and let sit for 5 minutes. (The residual heat will increase the internal temperature to about 130ºF, a perfect medium, and also create steam to keep the fish from drying out.) Uncover and serve with the lemon wedges and pesto.