Heat 6 tbsp. of the butter in a large skillet until it is hot and foaming. Add the shrimp, salt, and black pepper and cook over high heat, stirring often, for 4–7 minutes, until the shrimp are pink and cooked through.
Remove the skillet from the stove and use a slotted spoon to transfer the cooked shrimp to the bowl of a food processor fitted with a steel blade. Return the skillet to the stove and add the sherry, lemon juice, and cayenne. Cook over high heat until the liquid in the skillet is reduced to approximately 3 tbsp. and is quite syrupy. Immediately add this to the shrimp in the food processor and process until the shrimp are thoroughly puréed. With the motor running, add the remaining butter, a few pieces at a time, and process until thoroughly blended. Turn the food processor off and carefully taste the shrimp paste for seasoning, adding more salt, black pepper, sherry, lemon juice, or cayenne as needed. Transfer the shrimp paste to a ceramic crock and allow to cool completely. If not using right away, cover the shrimp paste and refrigerate for up to 1 week.