Trim excess fat from chicken thighs and season with salt and pepper. Heat oil in a large lidded skillet over medium-high flame. Add chicken thighs, skin side down, and cook until golden brown, 4 to 5 minutes.

Turn chicken thighs and reduce heat to medium. Cook for about 4 minutes, then partially cover pan and cook chicken, turning occasionally, until cooked through. Depending on thickness of chicken, this could take 10 to 20 minutes. An instant read thermometer should register at least 160ºF when inserted in the thickest part of the thigh.

Transfer chicken to a plate and set aside. Pour off all but 2 tablespoons of fat in the pan and add the shallots and tarragon. Cook, stirring occasionally, until shallot is softened, 2 to 3 minutes.

Add the wine and broth to the pan, scraping up any browned bits. Cook for a minute or so, until the liquid is slightly reduced.

Whisk in the half & half and mustard, and cook, stirring occasionally, for about 2 minutes. Taste the sauce. If the mustard is too subtle (but remember, you don’t want it to take over), whisk in another teaspoon or two.

Return chicken to pan, turning to coat with the sauce, and cook until it is just heated through, 2 to 3 minutes.

Transfer chicken to a platter or plate individual servings. Spoon sauce over chicken and serve.