Position a rack in the center of the oven and pre-heat oven to 350 F.
Heat oil in a 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 tablespoon of fat from the pot and then add the leek(s) 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 stock, season lightly with salt and pepper, and bring to a boil.
Arrange the chicken in the pot, skin side up, and return to a boil, and then transfer the pot to the oven to braise, uncovered, until the chicken cooks through, about 25 minutes.
Take the pot out of the oven, and 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 to 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 tablespoon 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.