Heat olive oil in a large skillet over medium heat.
Add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for 30 seconds until fragrant.
Stir in the shredded chicken, ground cumin, chili powder, smoked paprika, salt, and black pepper. Cook for 2-3 minutes until well combined and heated through.
In a medium bowl, whisk together the green enchilada sauce, sour cream, and milk until smooth.
Reduce heat to low and pour half of the creamy sauce mixture into the skillet with the chicken. Stir to combine.
Remove the skillet from heat. Spoon about 1/4 cup of the chicken mixture onto each tortilla, roll them up, and place seam-side down back into the skillet in a single layer.
Pour the remaining creamy sauce evenly over the rolled tortillas in the skillet.
Sprinkle the shredded Monterey Jack and cheddar cheeses evenly on top.
Cover the skillet with a lid and cook over low heat for 10-12 minutes until the cheese is melted and the enchiladas are heated through.
Remove from heat and garnish with chopped fresh cilantro before serving.