Set oven control to broil. Brush both sides of tuna steaks with 1 teaspoon oil. Place on rack in broiler pan. Broil with tops 4 inches from heat 8 to 10 minutes, turning once and sprinkling with 1/4 teaspoon salt, until tuna flakes easily with fork.
Meanwhile, in food processor bowl with metal blade, place remaining ingredients except broth and cheese. Cover and process about 10 seconds or until finely chopped. With processor running, slowly pour in broth and continue processing until almost smooth. Stir in cheese. Serve with tuna.