Preheat the oven to 375°F (190°C).
Heat olive oil in a skillet over medium heat. Add the onions and garlic, and sauté until softened, about 3-4 minutes.
Add shredded chicken, chicken broth, and green enchilada sauce. Stir to combine and let simmer for 5 minutes. Season with salt and pepper to taste.
In a separate pan, lightly fry each tortilla for 30 seconds per side in a bit of oil, just to soften them (this step helps to avoid sogginess).
Mix sour cream with a tablespoon of green enchilada sauce and set aside.
Spread a bit of the chicken mixture on the bottom of a baking dish. Fill each tortilla with chicken mixture and roll up. Place seam-side down in the dish.
Once all tortillas are rolled, pour the remaining chicken mixture over the top, then sprinkle with shredded cheese.
Bake for 20-25 minutes, or until the cheese is bubbly and golden.
Top with sour cream sauce and fresh cilantro before serving.
