Season chicken generously with salt and pepper. Dust lightly with flour and shake off the excess.
Melt butter in a large, heavy-bottomed pot or skillet that's set over medium-high heat. Sauté chicken in batches if necessary, until well-browned and crisp on all sides. Set aside.
Add the whole shallots to the pot and sauté them in the butter and chicken fat until they begin to soften and caramelize, about 10 to 12 minutes.
Add wine to deglaze the pot, scraping up any of the browned bits with a spoon and then add mustard and tarragon.
Add chicken back to the pan, skin side up. Simmer covered over low heat for 30 minutes.
Remove lid and add in cherry tomatoes. Continue to simmer uncovered allowing the sauce to reduce a little. Season to taste.