Season the chicken thighs with salt and pepper. Heat oil in a 12" cast-iron or heavy skillet over medium-low heat.
Place the chicken thighs in the skillet skin sidedown and cook for 5 minutes until fat begins to render.
Turn the heat up to medium-high and rearrange the chicken pieces so that the fat gets under each piece. Continue to cook skin side down for another 10 minutes.
Turn chicken pieces over being careful not to tear the skin. Place skillet in oven for an additional 10-15 minutes until chicken reaches 165º. Set chicken pieces aside and keep warm.
For Sauce
Pour off all but 1 tablespoon fat from the skillet. (Be careful. The handle is hot!)
Add onions and sauté over medium-high heat until soft and translucent.
Add sliced mushrooms and garlic, cook for 3 minutes.
Sprinkle flour over and stir in. Cook and continue stirring for another minute. Slowly pour in broth, stirring to avoid lumps.
Add rosemary and thyme to the sauce. Reduce heat and simmer for 15-20 minutes.
Whisk in butter, salt, and pepper.
Return chicken to pan for 5 minutes to warm. Remove herb sprigs and serve.
Notes
*You could serve this with egg noodles, mashed potatoes, or it would go great with Cheesy Soft Polenta.