In a bowl, mix together the tuna, mayonnaise, pesto, celery, shallot, cornichons, parsley, salt, and pepper.
Place the bagel halves on a baking sheet.
Preheat the oven to 375 F (190 C) and cook for 10 minutes, until the bagels are lightly browned.
Spread the tuna mixture over the bagels and sprinkle with parmesan cheese.
Return the bagels to the oven and cook for another 8 minutes, or until everything is hot.
Serve very hot.