Add the oil and butter to a 10 inch skillet.
Add the onions to the skillet along with a 1/2 teaspoon of the salt and 1/2 teaspoon of sugar. Cook for 20 minutes over medium low heat, stirring often.
Add the garlic and cook for 2 minutes longer then remove the onions from the skillet.
Place the seasoned chicken in the skillet skin side down. Cook for 4-5 minutes until browned then turn over and cook another 4-5 minutes on the other side.
Remove the thighs to a plate and cover with foil.
Add the mushrooms along with the other 1/2 teaspoon salt and 1 teaspoon of the thyme.
Cook until mushrooms have browned, stirring, about 5-7 minutes. Sprinkle the flour over the mushrooms, stir and cook for 1 minute longer.
Slowly pour in the chicken broth, stir well to get everything off the bottom of the skillet.
Place the chicken thighs back in the skillet, spooning some of the broth over the tops, cover and cook on medium-low heat for 25-30 minutes until the chicken is no longer pink inside.