In a large mixing bowl, whisk together the eggs, mayonnaise, flour, chopped dill, salt, black pepper, and lemon zest until smooth.
Add the shredded chicken and shredded mozzarella to the bowl and stir until the chicken is evenly coated in the mixture.
Cover the bowl and refrigerate the mixture for at least 30 minutes to firm up so patties hold their shape.
Use a trigger-release ice cream scoop or a 1/4-cup measure to portion the mixture into 12–15 portions, then form each portion into a 1/2-inch-thick patty.
Place the Panko breadcrumbs on a plate and press both sides of each patty into the crumbs to coat.
Heat a large non-stick skillet over medium heat and add 1 tablespoon of olive oil. Add half of the patties and sauté for 3–4 minutes per side, until golden brown; add more oil as needed and repeat with remaining patties.
When cooked, transfer patties off the heat, sprinkle with salt, and squeeze fresh lemon juice over them. Serve warm.