Pull off and discard turkey skin; rinse thighs and pat dry.
In a 4- to 5-quart nonstick pan over medium-high heat, melt butter. Add turkey thighs in a single layer and turn often until browned on all sides, 4 to 5 minutes total. Transfer turkey to a rimmed plate.
Meanwhile, peel garlic cloves. Add to butter in pan; stir often until lightly browned, 2 to 3 minutes. Add mushrooms, celery, and chilies; stir often until vegetables are lightly browned, 8 to 10 minutes. Add turkey and any accumulated juices, broth, and wine. Bring to a boil; cover, reduce heat, and simmer until turkey is very tender when pierced, 1 1/4 to 1 1/2 hours. Remove from heat and return turkey to rimmed plate. When turkey is cool enough to handle, pull meat from bones, tearing it into bite-size pieces; discard bones. Return meat to pan and stir in 1/2 cup minced cilantro.
Meanwhile, in a bowl, mix flour, baking soda, and 1/2 teaspoon salt. Add egg and milk; stir with a fork just until dough is evenly moistened and holds together.
Return turkey mixture to high heat; when boiling, reduce heat to simmering. Gently drop 2-tablespoon scoops of dough into pan in a single layer. Cover and simmer, without stirring, until dumplings are no longer moist in the center of the thickest part (cut to test), about 10 minutes.
Spoon turkey and dumplings into wide bowls. Sprinkle evenly with remaining 1/4 cup minced cilantro. Add salt to taste.