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Step 1
Pat the chicken thighs dry with a paper towel. Season generously on both sides with salt and pepper. Sprinkle the flour on both sides, patting off any excess.
Step 2
Pour the oil into a large (12-inch) saute or braising pan and place over medium-high heat for a minute or two. When the oil begins to shimmer, arrange the chicken in the pan, skin-side down.
Step 3
Cook the chicken until the skin is golden brown, 6-8 minutes. Turn the pieces over and cook another 2 minutes. Remove the chicken to a plate.
Step 4
Stir the shallots and bell pepper into the fat in the pan. Cook 2-3 minutes, stirring frequently, until softened. Add the mushrooms and 1 teaspoon salt. Cook, scraping up any browned bits from the bottom of the pan, until the mushrooms begin to release their liquid, 2-3 minutes.
Step 5
Add the garlic, tomato paste, balsamic vinegar and 1 tablespoon of the rosemary to the pan. Cook one minute, stirring to disperse the tomato paste into the cooking oil. Arrange the chicken pieces skin side up over the vegetables. Pour in the chicken broth and bring to a simmer.
Step 6
Cover the pan and cook 25 minutes. Adjust the heat as needed to maintain a slow bubble without boiling. Turn the chicken over and cook, uncovered, 10 more minutes. The chicken should be very tender when prodded with a fork.
Step 7
Taste the sauce for seasoning, adding more salt if needed. Sprinkle the chicken with the remaining tablespoon chopped rosemary and serve.