Heat olive oil. Sauté garlic and onions until onions are translucent and garlic is soft. Add spices to the skillet.
Add the pulled chicken to the skillet and stir. Pour 1 and 1/4 cups of enchilada sauce. Stir and set aside.
Cut off top and bottom of zucchini. Use a peeler to slice thin strips of zucchini. Pat strips dry with paper towel.
On a clean surface, lay down three over-lapping zucchini strips and add chicken mixture onto strips. Roll tightly and place in baking dish.
Place remainder of enchilada sauce over the roll ups and sprinkle with dairy free cheese. Bake in the oven until cheese is melted and zucchini is soft.