Prepare the sauce: Heat olive oil in a small saucepan over medium heat. Add onion and cook until soft, about 5 minutes. Add garlic and cook for 1 minute more.
Stir in tomato sauce, chili powder, cumin, and oregano. Add water and bring to a simmer. Cook for 5 minutes, stirring occasionally. Season with salt and pepper. Remove from heat.
Prepare the filling: In a bowl, mix the shredded chicken with 1/2 cup of the Monterey Jack cheese.
Preheat your oven to 375 degrees Fahrenheit (190 degrees Celsius). Lightly grease a 9×13 inch baking dish.
Soften tortillas: Heat vegetable oil in a small skillet over medium heat. Quickly dip each corn tortilla in the hot oil for about 10 seconds per side to make it pliable. Do not let them get crisp.
Assemble the enchiladas: Dip each softened tortilla into the red sauce, coating both sides. Place about 1/4 cup of the chicken mixture down the center of the tortilla. Roll it up tightly and place it seam-side down in the prepared baking dish. Repeat with remaining tortillas and filling.
Pour any remaining sauce over the rolled enchiladas. Sprinkle the remaining 1/2 cup Monterey Jack cheese and the 1/2 cup cheddar cheese over the top.
Bake for 20 to 25 minutes, or until the cheese is melted and bubbly and the sauce is hot throughout.
Notes
For a faster preparation, use store-bought rotisserie chicken for the filling.
If you prefer a thinner sauce, add a little more water while simmering.
Serve immediately with sour cream, chopped cilantro, or sliced avocado.