Trim the chicken thighs of excess fat. Season generously with salt and pepper and half the dried tarragon, pressing lightly to adhere seasonings to chicken. Dredge chicken in flour, shaking off excess.

Heat 2 tablespoons of oil and 2 tablespoons of butter in a deep lidded skillet large enough to hold the chicken in a single layer over medium-high flame, swirling the pan to combine the oil and butter. Brown the chicken in two batches to avoid overcrowding. Place 3 pieces skin side down in the pan and cook for 5 to 7 minutes, until golden brown. Turn and cook on the second side for 3 to 4 minutes. Transfer chicken to a plate and brown the second batch, then transfer that to the plate.

Pour off all but 2 tablespoons of fat, reduce heat to medium and add shallots to pan. Drizzle in additional oil, if needed, and sweat shallots, stirring frequently, until beginning to soften, about 3 minutes. Add remaining tarragon and garlic, and cook, stirring constantly, until fragrant, about 45 seconds.

Transfer chicken to a serving platter and tent with foil. Slice remaining 2 tablespoons of butter into pan, stirring to melt and combine into sauce. Cook for about 2 minutes, then transfer sauce to a serving bowl. Serve, encouraging guests to spoon shallot vinegar sauce over individual pieces of chicken. Or if you're not being fancy, plate it that way.