Lightly salt and pepper your chicken breasts, then dip them into your flour of choice. Cover both sides with flour, shaking off the excess. Save 1 tbsp of the flour for later.
In a large pan (emphasis on large!) over medium heat, add 1 tbsp of olive oil and 1 tbsp of butter or 2 tbsp of ghee. When shimmering, add flour coated chicken and fry until golden brown on both sides, about 3-4 minutes per side. Transfer to a plate and cover with foil.
In the same skillet with the drippings from the chicken, add 1 more tbsp of butter or ghee and melt. Add mushrooms for 3-4 minutes until browned on both sides The major tip here is to not crowd your mushrooms or they will steam. Give them some space in a single layer and they will get nicely browned.
Add your shallots and garlic to the mushrooms, scraping up any bits from the bottom. Mix in the 1tbsp of flour. Pour in the chicken broth & vinegar and simmer over medium heat for about 5-7 minutes, until reduced by half and thickened.
Add the heavy cream, mustard, and thyme. Return the chicken to the sauce mixture and simmer another 2-3 minutes to thicken. Top with parsley and serve!
I like to serve this with cauliflower rice or pureed cauliflower, and green beans.
Notes
I like to use ghee instead of butter here since it can reach a higher smoke point, but combining olive oil and butter helps to not burn the butter at a higher heat.