Transfer the red pepper juice to a medium saucepan. Add the butter, salt and pepper and cook over moderately high heat until the butter melts. Add the flour all at once and beat with a wooden spoon until the dough pulls away from the side of the pan and forms a smooth, shiny ball on the spoon. Scrape the dough into a bowl. Using an electric mixer at medium speed, beat in the eggs, 1 at a time, beating well between additions. Beat in the Pecorino Romano and cayenne.