The moment the butterâ€™s foaming subsides, slip the breasts into the pan and cook, partially covered or covered with a splatter shield, until browned, about 8 minutes, turning once but making sure several times that the breasts are not sticking. Transfer to a plate and set aside.
Melt the remaining tablespoon of butter in the skillet, then add the shallots. Cook, stirring often, until softened, about 2 minutes.
Add the mushrooms and cook, stirring once in a while, just until they start to give off their liquid, about 2 minutes. Stir in the broth, thyme, and mustard, then return the breasts to the pan with any accumulated juices.
Cover and cook until a meat thermometer inserted into the center of a breast registers 165Â°F (our preference) or 170Â°F (the USDA recommendation), about 10 minutes. Season with salt to taste before serving.