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Zest the lemon, set zest aside.
Prep the artichokes. Fill a med bowl with 4 cups cold water and squeeze with the juice from the lemon. Because artichokes will easily brown, work on one at a time as quickly as possible. Snap off all the hard out leaves using your fingers ( Save for later) With a sharp paring knife, trim off all the outer green skin around the base and stem, keeping the stem in tact. Remove the yellowish soft leaves and using a spoon, gently scrape away the white or purple silk like threads. Immediately place in the lemon water, to prevent browning. Slice just before placing in the skillet.
Preheat oven to 425F. Pat chicken dry and season with kosher salt and pepper. Heat 1 T butter and 1 T olive oil in a heavy bottom oven proof skillet over med high heat. Place chicken skin side down and sear for until golden, about 5 minutes. Turn and brown undersides until golden, about 7-8 minutes. Remove chicken and set aside. Drain some of the chicken fat, leaving about 1 tablespoon oil in the pan. Add 1 tablespoon more butter. Add leeks, sliced artichokes, sliced potatoes and fennel seeds. Saute on med heat for 8-10 minutes, stirring frequently, until they just begin to soften. Season with a generous pinch of kosher salt and pepper.
Deglaze the pan with vermouth (or white wine). Reduce by half, about 4 minutes. Add stock, mustard and fennel seeds, half and half, half of the fresh tarragon and 1/2 of the zest. Give a stir, bring to a simmer, and taste broth adjusting salt and pepper. Nestle the chicken pieces into the stew with skin sides up, and roast in a hot 425 F oven for 20-30 minutes, until golden, and thigh internal temp is 170F, or breast internal temp is 160F. If needed, broil for a couple minutes to get skin extra crispy.
Garnish with remaining lemon zest and fresh tarragon before serving.