Preheat oven to 400°F.
Season chicken thighs with 3/4 teaspoon salt and pepper. Sear, skin-side down, in a 10 or 12 inch cast iron skillet over medium heat until the skin is golden-brown and crispy, 7 to 10 minutes. Transfer the chicken to a holding plate.
If there is not at least 1 tablespoon of chicken fat in the skillet, add enough olive oil to get to 1 tablespoon total. Add the shallot and cook for 2 minutes, stirring constantly. Add the garlic and cook for an additional 30 seconds. Add the mushrooms and cook for 2 minutes, then stir in 1/4 cup dry white wine and the remaining 1/4 teaspoon salt and cook until it almost evaporates.
Return the chicken thighs to the skillet, trying not to cover any mushrooms if possible. Cook for 20 to 25 minutes, until an internal thermometer stuck into a chicken thigh reads 165°F.
Stir the butter and half of the parsley into the mushrooms, then garnish with the remaining parsley.