Combine the tomatillos, onion, serrano (including the seeds), garlic, cilantro stems and 1 teaspoon salt in a medium saucepan. Add enough water to just cover the vegetables and bring to a boil over medium-high heat. Reduce the heat and simmer until the tomatillos are tender and slightly darkened, about 15 minutes. Let cool slightly.
Transfer the vegetables to a blender with a slotted spoon; reserve the cooking water. Add the cilantro leaves and puree until smooth, adding a few tablespoons of the reserved cooking water if the salsa is too thick. Let sit 30 minutes to let the flavors develop. Add lime juice and salt to taste.