Place the cooked chicken cubes, eggs, kosher salt, black pepper, garlic powder, dill and cayenne in your food processor's bowl. Process into a uniform mixture.
Heat 2 tablespoons of the the olive oil in a large skillet over medium heat.
Drop the mixture into the hot oil, measuring 1/4 cup per patty and fitting 4 patties into the skillet. Gently flatten them to 1/2-inch thick with the back of a spoon.
Cook the chicken patties until crispy and golden brown, 3-4 minutes per side. Transfer to a plate and cover with foil to keep warm. Add the remaining olive oil to the skillet and cook the remaining patties. Serve immediately.