Roast poblanos directly over the flame of a gas-stove burner or under a broiler, turning often with tongs, until charred on all sides. Place in a bowl, cover with plastic wrap, and let stand for 15 minutes. When cool enough to handle, rub with a paper towel to remove skins, being careful not to tear flesh. Cut a slit down the side of each poblano from stem to tip. Remove seeds, leaving stem intact.
Meanwhile, place chicken and garlic in a saucepan, and add enough water to cover. Bring to a boil, then reduce heat, and simmer until chicken has cooked through, about 15 minutes. Transfer chicken to a large bowl, reserving cooking liquid and garlic. Cover chicken loosely with plastic wrap to prevent it from drying out. Let cool slightly, then shred it into 1/2-inch pieces, and cover.
Add tomatillos to reserved cooking liquid. Bring to a boil. Reduce heat and simmer until tender, about 5 minutes. Drain, and puree tomatillos and garlic in a blender. Add cilantro, lime juice, and 1 1/2 teaspoons salt, and puree. Spread half the sauce in a 9-by-13-inch baking dish.
Preheat oven to 375 degrees. Heat oil in a skillet over medium heat. Add corn, and cook until tender, about 3 minutes. Let cool slightly. Toss corn with chicken, remaining sauce, half the goat cheese, and the pepper. Mound 1/2 cup mixture into each poblano. Place in prepared dish, dot with remaining goat cheese, and sprinkle with remaining 1/4 teaspoon salt. Cover, and bake until heated through, about 25 minutes. Uncover, and let stand 5 minutes. Drizzle with sauce from dish, and garnish with cilantro sprigs.