Pat the chicken thighs dry and season both sides with salt and pepper.
In a large skillet, heat the oil over medium-high heat. Add in the chicken and cook on both sides until browned, about 5 minutes per side. Remove from the skillet and set aside.
Add in the butter, onion, carrots, and mushrooms, cooking until tender, about 7-10 minutes. Sprinkle over the flour and stir until well combined and no streaks of white flour remain. Stir in the wine, scraping up the browned bits from the bottom of the pan. Simmer, until the wine has reduced by half.
Stir in the broth, cream, and thyme. Add the chicken back to the skillet, bring the sauce to a simmer, and cook until the chicken is cooked through, about 15 minutes. Garnish with parsley and serve.
Wine: you can use additional chicken broth in place of the wine.
Cream: you can also use half and half in place of the cream.