Preheat oven to 350º. In large skillet over medium heat, heat oil. Add onion and season with salt. Cook until soft, 5 minutes, then add garlic, cumin, and chili powder and stir until combined. Add shredded chicken and 1 cup enchilada sauce and stir until saucy.
On a cutting board, use a Y-shaped vegetable peeler to make thin slices of zucchini. Lay out three, slightly overlapping, and place a spoonful of chicken mixture on top. Roll up and transfer to a baking dish. Repeat with remaining zucchini and chicken mixture.
Spoon remaining ⅓ cup enchilada sauce over zucchini enchiladas and sprinkle with cheddar and Monterey Jack.