Preheat oven to 475°. Pour bread crumbs, flour, and whites into 3 shallow bowls. Stir 1 tbsp. salt and the cayenne into bread crumbs and 1 tsp. salt into flour.
Pour 1/4 cup olive oil into a large rimmed baking pan and spread to cover bottom. Dredge shrimp in flour, shaking off excess; dip in egg whites, letting excess drip off; then roll in bread crumbs. Press each shrimp in oil, then turn to coat other side, adding more oil to pan if necessary to coat all shrimp. Set slightly apart in pan.
Bake shrimp until browned on the bottom, about 8 minutes; then turn over and bake until golden brown all over and opaque but still moist-looking in center of thickest part, 5 to 8 minutes longer.
Mix arugula with lemon juice, remaining 1 1/2 tbsp. olive oil, and salt and pepper to taste. Serve shrimp with arugula and Lemony Mayo.
Make ahead: Bread the shrimp earlier in the day and chill, uncovered (the coating dries out and the shrimp get crisper).
Wine Pick: White with the First Course. Spottswoode Sauvignon Blanc 2007 (Napa and Sonoma Counties; $32). Bright, complex citrus and stone fruit, with a rich core from a little Semillon blended in (the traditional white blend in Bordeaux).
Note: Nutritional analysis is per serving with mayo.