In a shallow dish, whisk together the flour, salt, pepper, paprika, and cayenne. In another shallow dish, whisk together the egg and water. Dredge each steak in the flour mixture, then dip in the egg mixture, then dredge again in the flour mixture.
Heat 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium heat. Add two of the steaks to the skillet and cook 3-4 minutes until crisp and golden brown, then flip and cook an additional 3-4 minutes until cooked through and golden. Remove the steaks to a serving plate and cover with foil to keep warm. Add the remaining olive oil and butter to the skillet, and cook the remaining two steaks until golden. Cover with foil while preparing the gravy.
To make the gravy, add the butter to the skillet with the browned bits from the steaks. Add the flour and whisk to form a paste. Cook 2-3 minutes, until golden. Gradually pour in the chicken broth, whisking constantly to prevent lumps. Add the milk, salt, and pepper, and cook until bubbling and thickened. To serve, spoon gravy over the steaks.