Chill the bowl and blade of a food processor. Trim the ends and sides of 8 sole fillets to square them off and lightly score the skin side of each fillet (this will help prevent them from curling while cooking). Dice the trim and remaining 4 fillets and transfer to the chilled food processor with the egg, 1¼ teaspoons salt, ¼ teaspoon pepper, and ½ teaspoon piment d’Espelette, and pulse until well combined. Scrape the sides of the bowl with a spatula, and with the machine running, stream in the heavy cream to make a smooth mousse. Pass the mousse through a fine-meshed drum sieve, transfer to a piping bag with a ½-inch-diameter tip, and reserve, chilled.