In a bowl, combine all ingredients for the salad. Keep refrigerated until ready to serve.
For the Aioli: In food processor combine first 5 ingredients. Turn on processor and slowly add the oils for emulsification. Add the herbs, and salt and pepper. This sauce is enough aioli for 10 sandwiches.
Spread aioli on 1 slice of bread. Top with tuna salad, sliced red onion, cheese, and the other slice of bread. Grill on panini press for 3 to 5 minutes.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.