1 cup (4 oz.) shelled pistachios (not dyed red)
1/2 cup packed fresh basil leaves
2 tablespoons minced shallot
1/4 teaspoon salt
1/4 teaspoon black pepper
6 1/2 tablespoons unsalted butter, softened
6 6-oz. arctic char fillets (1 to 1 1/4 inches thick) with skin
1/2 tablespoon vegetable oil
Coarsely chop pistachios in a food processor. Add basil, shallots, salt, pepper, and 6 tablespoons butter and purée until mixture forms a paste. Preheat oven to 400°F. Remove any bones from fish with tweezers and pat fish dry. Heat oil and remaining 1/2 tablespoon butter in a 12-inch nonstick skillet over moderately high heat until foam subsides, then brown skin sides of fillets in 2 batches, 3 to 4 minutes per batch. Transfer fillets as browned, skin sides down, to a lightly oiled shallow baking pan (1 inch deep). Divide pistachio paste among fillet pieces and spread evenly in a 1/8-inch-thick layer over top of each. Bake fillets in middle of oven until just cooked through, 9-11 minutes.
Pistachio paste can be made 4 hours ahead and chilled, covered. Bring to room temperature before spreading.