Pat the chicken thighs dry with paper towels. Season evenly with kosher salt and black pepper.
Heat the olive oil and butter in a large Dutch oven or heavy-bottomed pot over medium-high heat. Place the chicken skin-side down and sear until golden brown, about 6 to 8 minutes. Flip and brown the other side for 4 minutes. Remove the chicken to a plate and set aside.
Lower the heat to medium. Add the quartered mushrooms, sliced carrots, and chopped onions to the pot. Sauté for 5 to 7 minutes until the vegetables are softened and the mushrooms are browned.
Add the minced garlic and cook for 1 minute until fragrant. Sprinkle the all-purpose flour over the vegetables and stir constantly for 1 minute.
Pour in the dry white wine to deglaze the pot, scraping up any browned bits from the bottom. Allow the wine to simmer and reduce by half, about 2 minutes.
Stir in the chicken broth, heavy cream, and thyme sprigs. Return the chicken thighs to the pot, nestling them into the sauce.
Bring the sauce to a gentle simmer. Reduce the heat to low, leave uncovered, and cook for 25 to 30 minutes until the chicken reaches an internal temperature of 165°F and the sauce thickens.
Remove from heat, discard the thyme stems, and garnish with fresh chopped parsley before serving.