Sprinkle chicken evenly with 1/4 teaspoon each salt and pepper. Place flour in a shallow dish. Dredge the chicken, 1 cutlet at a time, in the flour, pressing to adhere and shaking off excess flour. Place the coated chicken on a large plate. Discard any remaining flour.
Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add the chicken; cook until golden brown on both sides and a thermometer inserted into the thickest portion registers 165°F, 4 to 5 minutes per side. Transfer to a plate. Do not wipe the pan clean.
Return the skillet to medium-high heat. Add mushrooms and the remaining 2 tablespoons oil. Spread the mushrooms into an even layer; cook, undisturbed, until golden, about 5 minutes. Stir the mushrooms; add shallot, garlic and thyme. Cook, stirring often, until fragrant and the shallot is translucent, 2 to 3 minutes.
Stir in vinegar; cook until the mushrooms have absorbed the liquid, about 1 minute. Add broth; bring to a boil over medium-high heat. Reduce heat to maintain a simmer. Add cream cheese, mustard and the remaining 1/4 teaspoon each salt and pepper, stirring until melted.
Return the chicken to the pan; reduce heat to medium-low and cook until warmed through, 1 to 2 minutes. Garnish with additional thyme leaves before serving, if desired.