Pour oil to a depth of 2 inches in a large deep skillet; heat over medium-high to 370°F. While oil heats, whisk together eggs, milk, and hot sauce in a shallow dish; add shrimp to the mixture, and toss to coat. Whisk together flour, cornmeal, salt, black pepper, and cayenne pepper in a separate shallow dish.

Remove 6 to 8 shrimp from egg mixture, shaking excess off; dredge in flour mixture. Place dredged shrimp in hot oil; fry until coating is golden and crisp, about 2 minutes. Remove with a slotted spoon to a baking sheet lined with paper towels. Repeat with remaining shrimp, bringing oil back to 370°F over medium-high heat before adding each batch of shrimp.

Serve fried shrimp with lemon wedges, cocktail sauce, and rémoulade if desired.