In a large mixing bowl, combine the shredded chicken, cheddar cheese, Monterey Jack cheese, chopped onion, chopped cilantro, cumin, salt, and black pepper. Mix thoroughly until all ingredients are evenly incorporated.

Make the Enchilada Sauce

Heat the vegetable oil in a saucepan over medium heat. Whisk in the flour and cook for 1-2 minutes, stirring constantly to form a roux.

Add chili powder, cumin, garlic powder, oregano, and salt. Stir well to combine and cook for another 30 seconds.

Whisk in the tomato paste until smooth, then gradually add the chicken broth while continuing to whisk to avoid lumps.

Bring the sauce to a simmer and let it cook for 8-10 minutes, stirring occasionally, until slightly thickened. Set aside.

Assemble the Enchiladas

Preheat oven to 375°F (190°C).

Heat a small amount of vegetable oil in a skillet over medium heat. Lightly fry each tortilla for about 10 seconds per side to soften. Drain on paper towels.

Spread 1/2 cup of enchilada sauce in the bottom of a 9x13-inch baking dish.

Take one tortilla at a time, place about 3 tablespoons of filling in the center, roll it tightly, and place seam side down in the prepared baking dish. Repeat with remaining tortillas and filling.

Pour remaining enchilada sauce evenly over the rolled tortillas.

Sprinkle the reserved cheddar and Monterey Jack cheese over the top.

Cover with foil and bake for 20 minutes. Remove foil and bake for another 10 minutes or until cheese is melted and bubbly.

Garnish with freshly chopped cilantro before serving, if desired.