If using frozen fish, thaw it in the refrigerator overnight and bring to room temperature before cooking.
Preheat your oven broiler to high and position a rack so the fish will be a few inches from the broiler element.
Pat the fish fillets dry with paper towels and season both sides with salt, pepper, and the Szeged Fish Rub.
Heat a heavy frying pan over medium-high heat and add enough olive oil to coat the pan.
Sear the fillets in the hot pan for about 3–4 minutes per side, until lightly browned and almost cooked through; turn carefully to keep the fillets intact.
Transfer the nearly cooked fillets to an ovenproof baking dish that fits under your broiler.
Spread about 1 tablespoon mayonnaise on top of each fillet, then sprinkle the grated Parmesan evenly so the tops are well covered.
Broil the fillets until the cheese is melted and browned, about 3–5 minutes—watch closely so they do not burn.
Remove from the oven and serve hot, optionally with lemon slices.