Place a rack closest to broiler. Heat broiler to high. In a medium bowl, combine mayonnaise, sriracha, oil (if using), and 4 teaspoons soy sauce.

Place salmon on a large foil-lined baking sheet; season all over with 1 teaspoon salt. Spoon 1 tablespoon spicy mayo on each fillet and rub all over. Broil salmon, watching closely, until tops start to brown and blister, 9 to 11 minutes.

Meanwhile, in a large bowl, toss rice, furikake, 1 tablespoon vinegar, 1 teaspoon sugar, and remaining 2 teaspoons soy sauce until combined. In a small bowl, toss cabbage, ¼ teaspoon salt, and remaining 2 tablespoons vinegar and ¼ teaspoon sugar until combined.

Transfer salmon to another medium bowl and add 2 tablespoons spicy mayo. Using a fork, break salmon into small pieces and toss until coated.

On a work surface, arrange a sheet of nori with a point facing you. Spoon ⅓ cup rice mixture into the center. Using wet fingers, pat rice into a square. Top with ¼ cup salmon mixture, spreading over rice. Layer with 4 slices cucumber, 2 tablespoons cabbage mixture, 1 tablespoon grated carrot, 2 slices avocado, and 2 tablespoons spicy mayo. Fold top corner of sheet down over the center. Repeat with bottom, left, and right corners to make a square pouch, enclosing fillings. Tightly wrap with plastic wrap. Repeat with remaining sheets and filling. Let sit 10 minutes.

Unwrap sushi sandwich and slice in half. Serve with remaining spicy mayo alongside.