Preheat oven to 375°F. Mist a 9-by-13-inch baking dish with cooking spray. Warm oil in a medium skillet over medium heat. Add onion and cook, stirring, until softened but not browned, 5 minutes. Add garlic and jalapeño and cook 2 minutes longer, stirring often. Remove from heat and let cool slightly.
In a large bowl, combine onion mixture, chicken, 1 cup cheese, 1/2 cup salsa and mild green chilies. Stir until well combined.
Pour broth into skillet and heat until almost simmering. Working one at a time, dip a corn tortilla into broth, taking care to work quickly so tortilla doesn't become too soft, then transfer to a clean workspace. Spoon about 2 Tbsp. of chicken mixture down center of tortilla. Roll tortilla around filling like a cigar, and transfer, seam side down, to prepared dish. Repeat with remaining tortillas and filling. Pour remaining salsa over top of enchiladas, spreading to coat. Sprinkle remaining cheese over top.
Bake enchiladas, uncovered, until cheese is melted and enchiladas are hot throughout, about 20 minutes. Just before serving, sprinkle with cilantro.