In these Caesar-salad bites, Parmesan crisps hold marinated anchovies with chopped lettuce.
Preheat oven to 375 degrees. Line each cup of an egg carton with aluminum foil; set aside. Line a baking sheet with a Silpat; warm in oven 5 minutes. Remove baking sheet from oven; place 12 heaping tablespoons Parmesan 1 1/4 inches apart on Silpat. Bake 6 to 7 minutes, until golden and bubbling. Remove baking sheet from oven; let stand 2 minutes.
Using an offset spatula, remove Parmesan crisps, and quickly place each in a prepared egg-carton cup to mold into a cupped shape. Let set until crisp, about 3 minutes. If crisps harden before they can be placed in the cups for molding, return to baking sheet and place in oven 2 minutes to soften. Repeat with remaining Parmesan. To store overnight, place in airtight container at room temperature.
In a medium bowl, combine lettuce, chives, and shallots. In a separate bowl, whisk oil and lemon juice; season with salt and pepper. Toss dressing with salad.
Fill each Parmesan crisp with a heaping tablespoon salad, and top with an anchovy half. Serve immediately.