Pat the chicken breasts dry with paper towels. Season both sides to taste with salt and pepper.
Heat the oil in a large skillet over medium-high heat. Add the chicken and cook undisturbed until browned on one side, about 5 minutes.
Flip the chicken over, reduce the heat to medium, and cover the skillet. Cook about 10-15 minutes more, or until an instant-read thermometer inserted in the thickest part registers 160–165°F. Transfer to a plate and tent loosely with foil to keep warm.
Add the mushrooms and butter to the skillet. Cook, stirring frequently until the mushrooms have released their juices and are starting to brown, 4-5 minutes.
Add the shallot and cook until it begins to soften, about 1 minute.
Add the wine and use a wooden spoon to scrape up all the browned bits from the bottom of the pan. Let it simmer over medium heat until the liquid is reduced by about half and no longer smells strongly of alcohol, 3-5 minutes.
Take the skillet off the heat and whisk in the Boursin cheese until melted and smooth, then stir in the parsley. Season to taste with salt and pepper.
Return the chicken and any accumulated juices to the pan, spoon the sauce over the top, and let it warm through for 1-2 minutes over low heat before serving.