Preheat oven to 400ºF with rack set in middle position.
Combine ¾ cup salsa verde and ¼ cup sour cream. Add chicken, kale and 1 cup cheese; season with salt and pepper to taste.
In a separate bowl, whisk together 1 cup salsa verde, remaining ½ cup sour cream, and 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 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, bubbly and starting to brown, 15–20 minutes. For a golden brown top, turn oven to broil and broil for 2 minutes (optional).
Top enchiladas with cilantro and serve with remaining salsa verde and additional sour cream, if desired.