Place 1/3 of the chicken pieces into the bowl of a food processor and pulse several times until broken down into small pieces; do not overprocess. Transfer chicken to a mixing bowl. Repeat 2 more times with remaining chicken.
Place celery, onion, carrots, garlic, and rosemary into the food processor and pulse several times until vegetables are 1/4 inch in size. Add to the mixing bowl with the chicken.
Add egg, almond flour, mayonnaise, salt, and pepper to the bowl with the chicken mixture. Stir until evenly combined.
Pour 1/4 inch of avocado oil into a 12-inch nonstick skillet over medium-low heat. Heat until shimmering. Use a 1/4-cup measure to drop 6 portions of chicken mixture into the oil. Lightly press each mound into a 3-inch fritter. Cook until chicken is no longer pink and tops are browned, 2 to 3 minutes on each side.
Transfer cooked patties to a plate lined with paper towels. Repeat with the remaining mixture. Serve warm.