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Add the oil and butter to a 12 inch skillet over medium-low heat.
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, stirring often. Add the garlic and cook for 2 minutes longer then remove the onions from the skillet and cover to keep warm.
Place the seasoned chicken thighs 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 to the skillet along with the remaining tablespoon of butter, the remaining 1/2 teaspoon salt and the remaining 1 teaspoon of 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 while whisking to prevent any lumps, and get anything off the bottom of the skillet.
Place the onions and chicken thighs back in the skillet, skin side up. Spoon some of the broth over the tops, cover and cook on medium-low heat for 25-30 minutes until the chicken is cooked through.
Remove the chicken thighs to a serving platter. Pour the heavy cream in the skillet, whisking to combine with the sauce. Simmer for 2-3 minutes then pour the gravy over the chicken thighs and garnish with parsley.