Trim any excess fat from the chicken breasts and dice into roughly 1/3-inch pieces; place the diced chicken in a large mixing bowl.
Add 2 eggs, 1/3 cup mayonnaise, 1/3 cup flour (or chosen starch), shredded mozzarella, chopped dill, 1/2 tsp salt, and 1/8 tsp black pepper to the bowl and stir until evenly combined; cover and refrigerate 2 hours or overnight to firm up.
Heat a large non-stick skillet over medium heat and add 2 Tbsp oil.
Spoon a heaping tablespoon of the chicken mixture into the hot pan for each fritter, slightly flattening each with the back of the spoon to form patties.
Sauté uncovered 3–4 minutes on the first side until golden, then flip and cook about 3 minutes more, or until the exterior is golden brown and the chicken is cooked through; add more oil as needed and repeat with remaining batter.
To make the aioli, combine 1/3 cup mayonnaise, pressed garlic, 1/2 Tbsp lemon juice, 1/4 tsp salt, and 1/8 tsp black pepper in a small bowl and stir until smooth.
Serve the fritters warm with the aioli if desired.