Coat the chicken thighs in ¼ cup flour, shaking off the excess.
Heat 1 tablespoon butter and 2 tablespoon oil in a deep saute pan over medium-high heat. Sear the chicken on both sides until golden brown, about 5-6 minutes. Remove the chicken from the pan and set it aside.
Lower the heat to medium. In the same pan, add the onion and cook until it's deep brown and caramelized, about 20 minutes. If the pan gets too dry, add 1-2 tablespoon of water.
Add the garlic cloves and cook for 30 seconds or until aromatic. Add the tomato paste and caramelize for 1 minute.
Add the chicken back to the pan along with the sherry vinegar, chicken broth, mushrooms, thyme, and bay leaf. Cover and bring to a boil. Reduce the heat to low and continue to simmer for 10 minutes.
Remove the lid and transfer the chicken to a plate. Add the sugar to the sauce and increase the heat to medium-high. Continue to simmer the braising liquid until it's slightly thickened, about 10 minutes. Season the sauce with salt and pepper and pour it over the chicken. Serve immediately with crusty bread, mashed potatoes, or rice.