In a medium-size bowl, combine shallot, lemon juice, vinegar, and lime zest. Cover and let stand 30 minutes.
Whisk in the mayonnaise and the Piment d'Espelette chile pepper. Season with salt and pepper to taste.
Cover and refrigerate at least 2 hours.
Can be made 1 day ahead. Store in the refrigerator.
Piment d'Espelette Mayonnaise Sauce Recipe: https://whatscookingamerica.net/karencalanchini/piment-d-espelette-mayonnaise.htm