If scallops are quite large, cut them so the sizes are fairly uniform.
Rinse the scallops under cold water and pat them dry with paper towels. Remove the tough white side muscle if not already removed.
Heat about 2 to 3 inches of vegetable oil in a deep, heavy cast iron skillet, saucepan, or deep fryer to 350 F.
Line a baking pan or baking sheet with two or three thicknesses of paper towels and set aside.
In a bowl, combine flour, salt, and paprika.
In another bowl, whisk eggs with the milk.
Put the fine breadcrumbs in a third bowl.
Dip a few of the scallops in the seasoned flour mixture, then in the egg mixture, and then roll them in the breadcrumbs to coat. Repeat with the remaining scallops.
Once the oil has reached the temperature (350 F), add several scallops―about 6 to 10 depending on the pan size―to the hot oil and fry until golden brown, about 1 1/2 to 2 minutes. Take care not to overcook. They should be warm in the middle, but still moist.
Remove them from the oil with a metal slotted spoon and transfer them to the paper-towel-lined pan. Shake the pan gently for a few seconds so the scallops roll around and excess oil is absorbed.
Repeat with subsequent batches.
Serve immediately with your choice of cocktail sauce, remoulade sauce, or tartar sauce (see below).