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Adjust oven rack to middle position and heat oven to 300 F. Pat chicken dry with paper towels and season with salt and pepper. Heat oil in Dutch oven over medium-high heat until just smoking. Brown thighs, 5-6 minutes per side. Transfer to plate and discard skin.
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Pour off all but 1 tbsp fat from pot. Add onion, mushrooms and ½ tsp salt and cook, stirring occasionally, until vegetables are softened and beginning to brown, 6-8 minutes. Stir in garlic and thyme and cook until fragrant, about 30 seconds. Stir in flour and cook 1 minute. Slowly whisk in wine, scraping up any brown bits and smoothing out any lumps.
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Stir in broth, tomatoes and parmesan rind, if using, and bring to a simmer. Nestle chicken into pot, adding any accumulated juices, cover and transfer to oven. Cook until chicken registers 195 F, 35 to 40 minutes.
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Remove pot from oven and transfer chicken to serving platter. Discard cheese rind, if using. Stir sage into sauce and season with salt and pepper to taste. Spoon sauce over chicken and serve.