Season the chicken with ½ teaspoon salt and ¼ teaspoon pepper.
Heat 2 tablespoons of the oil in a skillet over medium-high heat. Cook the chicken in batches until golden brown, about 2 minutes per side. Transfer to plates.
Add the mushrooms and remaining tablespoon of oil to the pan and cook, tossing occasionally, until tender, 4 to 5 minutes.
Stir in the heavy cream, goat cheese, parsley, and ¼ teaspoon each salt and pepper. Serve over the chicken.