Preheat the oven to 350 degrees F (175 degrees C).
Heat oil in a large skillet over medium heat. Add onion and cook until soft, about 5 minutes. Add garlic, cumin, chili powder, and oregano. Stir until combined. Add shredded chicken, 1 cup enchilada sauce, and green chiles. Stir until well combined.
Slice zucchini into thin, wide sheets using a vegetable peeler or a mandoline. Lay out 3 zucchini slices, slightly overlapping, and place a spoonful of the chicken mixture on top. Roll up and transfer to a baking dish. Repeat with remaining zucchini and chicken mixture.
Spread remaining 1/3 cup enchilada sauce over zucchini enchiladas. Sprinkle with Cheddar-Jack cheese blend.
Bake in the preheated oven until cheese has melted, about 20 minutes. Garnish with fresh cilantro.
