Pat chicken dry and season with 1/2 teaspoon salt and pepper. Place flour in a shallow bowl and dredge chicken thighs to coat, shaking off excess. Discard remaining flour.
Heat butter and oil in a large skillet over medium-high heat until foamy.
Add the chicken and cook, flipping once until golden brown on both sides. About 7 minutes on the first side and about 5 minutes on the other side.
Transfer chicken to a plate. Then add the garlic, the shallot, and the remaining salt to the skillet. Cook for 1 minute.
Add the tomato paste and cook for 2 minutes.
Stir in the tomato, vinegar, and honey, scraping up any brown bits in the pan.
Stir in broth and bring to a simmer.
Add the chicken back to the pan and reduce the heat to low, cover and cook for 15 minutes.
Uncover and cook, flipping the chicken thighs occasionally, until the sauce is slightly thickened, for another 20-25 minutes.
Stir in the parsley and the tarragon.
To serve, spoon some of the tomatoes and sauce over each piece of chicken and serve with a side salad.