Preheat your oven to 100°C (210°F) and set a rack over a baking tray so you have a spot to keep the cooked rissoles warm.
In a large mixing bowl, combine the chicken mince, grated carrot, grated zucchini, egg, shredded cheese, breadcrumbs, parsley, salt, and pepper, stirring with a wooden spoon until everything is evenly mixed.
Wet your hands to stop the mixture from sticking, scoop just under ¼ cup at a time, and shape into patties about 6 cm wide and 1.5 cm thick.
Heat 1 tablespoon of oil in a large nonstick skillet over medium-high heat, cook 5 to 6 rissoles at a time for 3 to 4 minutes on each side until golden brown, then move them onto the rack in the oven to stay warm.
Add the remaining oil to the pan as needed and repeat the process until all the patties are cooked.
Serve the rissoles hot with ketchup for dipping.