In a blender, combine tomatillos, green chiles, garlic, chicken broth, cumin, salt, and lime juice. Blend until smooth.

Transfer the mixture to a saucepan and simmer over medium heat for 10 minutes.

Preheat the oven to 375°F (190°C).

In a skillet, heat olive oil over medium heat. Add diced onion and sauté until translucent.

In a mixing bowl, combine shredded chicken, half of the cheddar cheese, and half of the Monterey Jack cheese.

Warm the corn tortillas in a dry skillet until pliable. Fill each tortilla with the chicken mixture, roll them up, and place them seam-side down in a baking dish.

Pour the green sauce over the enchiladas and sprinkle with the remaining cheese.

Bake in the preheated oven for 20 minutes or until the cheese is melted and bubbly.