Slice each chicken breasts in half crosswise into 2 thin cutlets. Season each cutlet with salt and pepper. Add the flour to a large plate or baking tray. Coat both sides of each chicken cutlet in a thin layer of flour.
Heat 2 Tbsp. of oil in a large skillet over medium-high heat. Working in batches cook the cutlets the until golden brown, about 2 minutes. Turn over and cook on the other side just until the chicken is nearly cooked through, about 1-2 more minutes. Remove the cutlets from the pot and set on a clean plate.
In the same pot, add the butter followed by the mushrooms, shallot and garlic. Sauté for about 10 minutes, stirring frequently, until the mushrooms are golden brown and the garlic and shallot is fragrant. Season generously with salt.
Next, add the wine. Bring the wine to a simmer and reduce it by about half, stirring frequently and scraping up any caramelized bits from the bottom of the pan. Once the wine is reduced by half add the chicken stock, dijon mustard, cream, and fresh thyme.
Bring the sauce to a low simmer then add the chicken back into the pot. Let the chicken cook for another 5 minutes or so until it's cooked through and the sauce has thickened enough to coat the chicken. If it gets too thick add a little more chicken stock or cream. Remove the thyme stems, taste and season with salt & pepper as needed. Top with chopped parsley and serve immediately!