Pat the chicken cutlets dry with paper towels. Season both sides with salt and pepper. Dredge the chicken in the flour in a wide shallow bowl. Shake to remove excess flour.
Heat oil in a large skillet over medium high heat. Add chicken and cook until chicken is golden brown and fully cooked (about 4 to 5 minutes on each side). Transfer chicken to a plate and set aside (leave any juices in the skillet).
Add butter to the skillet and allow it to melt. Add sliced mushrooms and cook for 6 to 8 minutes, or until the mushrooms have a nice, browned sear and have released most of their moisture (do not stir very often to allow the mushrooms to sear). Remove mushrooms and place on a plate. Set aside.
Add minced garlic to the same (now empty) skillet and cook until fragrant (about 30 seconds), stirring constantly (add a splash of olive oil if necessary).
Add chicken broth and stir with a wooden spoon, scraping up any browned bits on the bottom of the skillet. Add heavy cream, mustard, parmesan, and thyme. Stir to combine. Bring to just a boil, then reduce heat and simmer for 3 to 4 minutes (stirring frequently), or until sauce is slightly thickened. Taste and season with salt and pepper, if desired.
Return the chicken and mushrooms to the skillet and coat the chicken with the sauce. Cook for an additional 1 to 2 minutes, or until chicken is heated through.