Pulse chicken breast in a food processor until finely chopped. Transfer to a bowl.

Combine celery, onion, and green onions in the food processor; pulse until finely chopped. Add to chicken in the bowl.

Stir mayonnaise, tarragon, salt, and black pepper into the chicken mixture in the bowl.

Chill chicken salad until flavors combine, at least 2 hours.

Top each pretzel cracker with a piece of provolone cheese and a heaping tablespoon of chicken salad. Garnish with a slice of cherry tomato.