Place chicken in a medium saucepan and add enough water to cover by at least 2 inches. Bring to a simmer over medium heat and cook until just cooked through, about 8 minutes. Remove from water; transfer to a plate. When cool enough to handle, shred into bite size pieces. You should have about 3 cups.
Meanwhile, preheat oven to broil with a rack in the position closest to the heating element. Arrange tomatillos, poblano, onion wedges, garlic, and jalapeno in a single layer on a rimmed baking sheet. Broil, flipping once halfway through, until blackened in places and beginning to soften, 8 to 10 minutes. Transfer to a blender; add cilantro, sour cream, 3/4 cup water, lime juice, and salt and puree until smooth.
Turn oven to 350 degrees. Spread 1 cup enchilada sauce evenly over the bottom of an 8-by-12-inch baking dish. In a medium bowl, toss chicken with 1 cup enchilada sauce. Place about 1/3 cup of chicken in the center of each tortilla. Roll up tortillas to enclose filling. Arrange enchiladas in a single layer in prepared baking dish, seam-side down. Pour remaining enchilada sauce over the top, then sprinkle evenly with cheese.
Bake until cheese is melted and sauce is bubbling, about 30 minutes. Let cool 10 minutes. Serve with cilantro leaves, chopped onion, and sliced jalapeno.
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