Heat the oil in large sauté pan over medium heat. Add the onion, season with salt and pepper and sauté until tender. Stir in the enchilada sauce and adobo sauce. Simmer over medium heat about 10 minutes, stirring often. Taste to adjust the seasonings. Reduce heat to low.
Combine the chicken and 1 cup of the cheese in a mixing bowl.
Preheat the oven to 350 degrees. F. Soften the tortillas by wrapping a stack of 5 at a time in a damp paper towel. Microwave for about 30 seconds. Repeat warming tortillas in batches. Fill each tortilla with about 3 T. chicken/cheese mixture and roll up.
Pour 1/3 cup of the enchilada sauce in a greased 13" x 9" baking dish. Arrange the enchiladas in one layer, seam side down. Cover the enchiladas with remaining sauce and sprinkle the remaining cheese over the top.
Bake for 25 minutes to 30 minutes until heated through!