Preheat oven to 400 degrees. After shredding the zucchini and squash, use your hands to squeeze any moisture from the vegetables.
In a large bowl, mix together the ground chicken, zucchini, squash, onions, bread crumbs, egg, cayenne pepper, and kosher salt. Using your hands, carefully create 4-5 chicken patties.
Add 1 tablespoons of olive oil to large cast-iron skillet or frying pan and allow the pan to get good and hot. Sear the chicken patties about 3-4 minutes per side or until golden brown.
Transfer the chicken patties to a baking sheet and bake for 8 minutes. Add a slice of cheese to the top of each patty and bake for two minutes longer.
Serve on your favorite burger buns with the toppings of your choice.