Preheat oven to 375°F (190°C).
Season chicken breasts generously with kosher salt.
Heat 2 tablespoons olive oil in a large skillet over medium-high heat. Sear chicken breasts until golden brown on both sides, about 5 minutes per side. Remove and set aside.
In the same skillet, melt butter. Add chopped serrano chiles and garlic; sauté until fragrant, about 2 minutes.
Whisk in flour and cook, stirring constantly, for 1 minute to form a roux.
Gradually whisk in chicken broth, scraping up any browned bits. Bring to a simmer and cook until thickened, about 3–5 minutes.
Add sour cream, cumin, cayenne, and tomatillos. Stir to combine and warm through gently, do not boil.
Remove chicken from the skillet, shred it with two forks, and return the shredded chicken to the sauce. Stir in half of the chopped cilantro.
Warm the corn tortillas by wrapping them in a damp kitchen towel and microwaving for 30 seconds to make them pliable.
Spread about 1 cup of the sauce evenly across the bottom of a large baking dish.
Fill each tortilla with 1/4 cup of the chicken mixture, roll tightly, and place seam side down in the baking dish.
Pour the remaining sauce over the rolled enchiladas, then sprinkle the shredded Monterey Jack cheese evenly on top.
Bake uncovered for 20–25 minutes, until cheese is melted and bubbly.
Garnish with remaining cilantro before serving.