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Trim the chicken thighs of extra skin and if using chicken breasts cut each breast in half. Arrange the chicken on a tray. Sprinkle the chicken on all sides with Kosher salt and let rest on the counter while you make the marinade.
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Peel the garlic, remove the green germ, and mince half of the garlic cloves. Place the minced garlic in the mixing bowl. Strip the thyme and tarragon leaves from their stems and finely chop. Add the herbs to the bowl. Add the lemon zest, mustard, honey, olive oil, ground pepper and 1/4 cup vermouth to the bowl and whisk together until incorporated.
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Cut the fennel bulb in half, remove the core and thinly slice the bulb, no more than a 1/4 inch. Reserve the fennel fronds and set aside.
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Peel each shallot then cut each bulb in half. Set aside with the fennel.
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Add the chicken to the marinade then use your clean hands and mix the chicken until each piece is thoroughly coated with the marinade. Add the sliced fennel, remaining garlic cloves, and shallots then mix them all together. Cover the bowl with plastic wrap and let marinate for 1 - 3 hours. Marinade the chicken in the refrigerator if longer than one hour.
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Preheat the oven to 425 F, 15 minutes before you want to bake the chicken.
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Spread the chicken, fennel and shallots over a large roasting pan that is just large enough to accommodate all the chicken without crowding the pan. You can slip the fennel under the chicken.
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Place the lemon wedges around the pieces of chicken, fennel fronds and the remaining 1/4 cup of vermouth to the roasting pan.
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Bake the chicken and vegetables for 20 minutes. Baste the chicken with pan juices and bake for 20 more minutes. Check to see if the chicken is done. The chicken is done when piercing the chicken with a fork, there is little resistance and the juices are clear. The internal temperature should be between 165F and 170F.
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When done, remove the chicken from the oven and let it rest for 5 minutes. Serve the chicken with the fennel, shallots and lemon, drizzled with remaining pan juices.