Sprinkle chicken evenly with ¼ teaspoon each of salt and pepper.
Place flour in a shallow dish. Dredge the chicken, 1 cutlet at a time, in the flour, pressing to coat and shaking off the excess.
Place the coated chicken on a large plate. Discard any remaining flour.
Heat 2 tablespoons of 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 oily pan clean.
Return the skillet to medium-high heat. Add the remaining 3 tablespoons of oil and then the mushrooms.
Spread the mushrooms into an even layer; cook, undisturbed, until golden, about 5 minutes.
Stir the mushrooms, then add the 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 the heat to maintain a simmer.
Add cream cheese, mustard, and the remaining ¼ teaspoon each of salt and pepper, stirring until melted.
Return the chicken to the pan; reduce the heat to medium-low and cook until warmed through, 1 to 2 minutes.
Garnish with additional thyme leaves before serving, if desired.