This classic French dish is typically made with a cut-up whole chicken. But using just one part of the bird, in this case thighs, ensures that the pieces are done at the same time. Serve this saucy braise with mashed potatoes.
Pat chicken dry and season with 1/2 teaspoon salt and pepper. Place flour in a shallow bowl and toss the chicken in it to coat, shaking off excess. (Discard the remaining flour.)
Heat butter and oil in a large skillet over medium-high heat until foamy. Add the chicken and cook, flipping once, until brown on both sides, 9 to 12 minutes total. Add garlic, shallot and the remaining pinch of salt; cook for 1 minute. Add tomato paste; cook for 1 minute. Stir in tomato, vinegar and honey, scraping up any browned bits. Stir in broth. Bring to a simmer.
Reduce heat to low, cover and cook for 15 minutes. Uncover and cook, flipping the chicken occasionally, until the sauce is slightly thickened, 20 to 25 minutes more. Stir in parsley and tarragon.