Preheat oven to 375ºF. Pat chicken dry with paper towels and sprinkle with flour, salt and pepper, on both sides.

Melt 2 tablespoons butter in skillet over medium-high heat until foaming. Brown chicken in two batches until golden brown and crisp, 2-3 minutes per side. Set aside.

If pan is dry, add last tablespoon of butter. Lower heat to medium and sauté shallots, stirring occasionally, for 10-12 minutes, until softened and starting to caramelize. Pour in vermouth and scrape bottom of the pan with a wooden spoon, as the wine bubbles, to deglaze and release all the tasty browned bits into the sauce. Let the sauce heat up for 30 seconds, then turn off heat, stir in the mustard, add tarragon sprigs, then chicken and tomatoes, tucking the tomatoes between the pieces of chicken. Cover pan tightly and cook in the oven for 30 minutes.

Remove lid and cook for an additional 20 minutes, until the sauce has thickened and the chicken is cooked through and the skin is crisped and browned (see notes below). Serve hot.