Preparation

• Position a rack in the center of the oven and heat the oven to 350°F.
• Heat the oil in an 8-quart Dutch oven or other heavy-duty pot over medium heat. Add the prosciutto and cook, stirring occasionally, until crisp, about 5 minutes. With a slotted spoon, transfer to a bowl and set aside. if the pan is dry, add a little more oil.
• Season the chicken lightly on all sides with salt and pepper. Working in batches, brown the chicken on both sides, about 12 minutes total. Transfer to a plate.
• Turn the heat down to medium low. Pour off all but 1 Tbs. fat from the pot and then add the leek and garlic. cook, stirring often, until beginning to soften, about 5 minutes. Add the flour and cook, stirring, for 1 minute. Add the reserved marinade and cook, scraping up any browned bits from the bottom of the pan, until the liquid thickens, about 2 minutes. Add the broth, season lightly with salt and pepper, and bring to a boil. Arrange the chicken in the pot skin side up, return to a boil, and transfer the pot to the oven to braise, uncovered, until the chicken cooks through, about 25 minutes.
• Take the pot out of the oven. Turn the broiler on high. Transfer the chicken, skin side up, to a foil-lined rimmed baking sheet.
• Simmer the sauce in the pot over medium-high heat, stirring occasionally, until thickened and reduced by about half, about 10 minutes. Lower the heat to medium and stir in the capers and peaches; cook until heated through. Stir in the butter until it melts, then stir in 1 Tbs. of the tarragon and season to taste with salt and pepper.
• Meanwhile, broil the chicken until the skin is crisp, about 3 minutes.
• Return the chicken to the pot or transfer it to a large platter and spoon the sauce over it. garnish with the prosciutto and the remaining tarragon leaves, and serve.