Mix egg and water together to make a light egg wash. Place breadcrumbs in a shallow dish.
Dip shrimp into the egg wash and toss into breadcrumbs.
Heat 2 tablespoons oil in a pan over medium heat. Add shrimp and fry until firm and opaque, 4 to 6 minutes. Remove shrimp and keep warm.
Rinse out the pan. Heat butter and remaining 2 tablespoons oil in the pan. Add lime juice; bring to a boil. Continue boiling for 1 minute. Return cooked shrimp to the pan and stir to coat with the lime sauce. Serve immediately.