Preheat oven to 400ºF (204ºC) with rack set in middle position.
In a large mixing bowl, combine 3 cups shredded chicken, 2 cups chopped kale, 1 cup salsa verde, 1 cup cheese, ½ cup sour cream, ¼ cup cilantro, 1 teaspoon cumin, ½ teaspoon salt and ¼ teaspoon pepper; mix to combine.
In a separate bowl, whisk together remaining 1 cup salsa verde, remaining ½ cup sour cream, and ¼ cup chicken broth; set aside.
Brush both sides of tortillas with olive oil; stack tortillas top of each other on a plate. Cover tortillas with a slightly damp paper towel and microwave for 30 seconds.
Spoon ⅓ cup chicken mixture onto a tortilla, slightly off-centered. Roll tortilla around filling and place, seam side down, in a 9x13-inch baking dish. Repeat filling and rolling with remaining tortillas and chicken mixture.
Spread reserved salsa verde-sour cream mixture over filled tortillas and sprinkle evenly with remaining ¾ cup cheese.
Bake enchiladas until cheese is melted and starting to brown, 30–32 minutes.
Let cool slightly before serving.