Place chicken breasts between plastic wrap and pound to an even 3/4-inch thickness. Pat dry with paper towels. Season both sides with salt and pepper. Lightly dredge in flour, shaking off excess.
Heat olive oil in a cast-iron skillet over medium-high heat until shimmering (375°F). Add chicken breasts and sear undisturbed for 5-6 minutes until golden brown. Flip and cook for 3-4 minutes more until internal temperature reaches 155°F. Transfer to a plate and tent with foil.
In the same skillet, add 1 tablespoon butter and mushrooms. Cook without stirring for 2 minutes to initiate browning, then stir occasionally for 5-7 minutes until deeply browned and no moisture remains in the pan.
Add shallots and sauté for 2 minutes until translucent. Add garlic and thyme, cooking for 30 seconds until fragrant.
Add white wine to deglaze, scraping up all browned bits with a wooden spoon. Reduce wine by 75% (about 3 minutes).
Add chicken stock and simmer for 3 minutes to reduce slightly. Reduce heat to medium-low.
Whisk in heavy cream and Dijon mustard. Simmer gently for 3-4 minutes until slightly thickened (should coat the back of a spoon).
Return chicken breasts to the skillet along with any accumulated juices. Simmer gently for 3-4 minutes until chicken reaches 165°F internal temperature and sauce further thickens.
Remove from heat. Swirl in remaining 1 tablespoon butter until melted. Taste sauce and adjust seasoning if necessary.
Transfer chicken to plates, spoon sauce over top, and garnish with fresh parsley.