Combine zucchini with coconut oil and butter in a medium skillet over medium heat. Add Parmesan cheese, chili powder, salt, and cayenne pepper. Stir until cheese melts.
Reduce heat to low and add eggs; stir quickly until fully mixed in. Increase heat back to medium and cook, stirring and flipping occasionally, until edges of the hash are lightly browned, about 15 minutes.
