Preheat the oven to 375°F with the rack in the center position.
Season chicken thighs on both sides with salt, paprika and black pepper.
Heat a large braising skillet or Dutch oven over medium heat. Add olive oil and swirl to coat.
Place the chicken skin-side down in a single layer and sear for 3 or 4 minutes until browned. Flip and sear the other side for another 3 or 4 minutes. Remove the chicken and set aside.
Add butter to the pan, then turn the heat down to medium-low.
Add the carrots and mushrooms, and cook for 5 to 7 minutes until browned and slightly softened. Stir in the garlic and red pepper flakes and cook for 1 minute.
Sprinkle in the flour and cook, stirring often, for 60 to 90 seconds to remove the raw flavor.
Deglaze the pan with half the wine, scraping up any browned bits. When the pan starts to look dry, add the remaining wine and chicken broth.
Stir in the pearl onions and thyme, then return the chicken to the pan and tuck in the bay leaves. Bring the mixture to a simmer, cover the pan and transfer it into the oven.
Roast for 30 minutes or until the chicken is very tender and reaches an internal temperature of 175°F.
Remove the lid and roast uncovered for 5 to 7 minutes until the chicken skin is crispy. Transfer the chicken to a plate.
Stir the heavy cream into the sauce and taste for seasoning, adding salt if needed.
Return the chicken to the pan and stir, then serve hot.