In a deep skillet, heat 3 tablespoons olive oil over medium heat. Add the garlic, jalapeño, and a pinch of salt, and cook until the garlic is softened, about 2 minutes. Add the tomatillos and ½ cup water and bring to a boil over high heat. Reduce the heat to medium, cover, and simmer until the tomatillos are softened and broken down, about 10 minutes.
Uncover and simmer over medium heat until thickened, about 4 minutes. Puree in a blender if desired, and return it to the skillet; keep warm. (Be sure to let the salsa cool slightly before pureeing. For a more rustic texture, you can skip this step.)
In a large bowl, toss the chicken with ½ cup of the salsa. Add half of the onion and cilantro and toss. Season with salt and pepper.
Using tongs, dip a tortilla in the salsa so it's completely submerged and warm it in the skillet until pliable, about 10 seconds. Transfer to a plate. Fill the tortilla with about ¼ cup of the chicken filling and roll it up to enclose. Transfer the tortilla, seam side down, to a platter or a shallow baking dish. Repeat with the remaining tortillas. Pour any leftover salsa over the enchiladas.
In a small bowl, toss together the remaining onion and cilantro with the cheese. Sprinkle mixture over the enchiladas and serve.