Preheat the oven to 425°. On a large rimmed baking sheet, toss the halved potatoes with the olive oil and season with salt and pepper. Roast the potatoes for about 35 minutes or until tender and browned. Set the potatoes aside. Reduce the oven temperature to 350°.
Meanwhile, in a large ovenproof skillet, melt the 3 tablespoons of butter. Season the chicken breast halves with salt and pepper and add to the skillet, skin side down, along with the thyme sprigs. Cook over moderately low heat until the skin is well browned, about 15 minutes. Turn and cook for 5 minutes longer, basting occasionally with the pan juices.
Transfer the chicken to the oven and roast for about 15 minutes, until cooked through. Add the chicken to the potatoes and keep warm. Reserve 3 tablespoons of the melted fat from the skillet.
Pour off the remaining fat from the skillet and discard the thyme sprigs. Add 1/2 cup of the wine to the skillet and boil over moderately high heat, scraping up the browned bits, until reduced by half, about 2 minutes. Add the chicken stock and boil until slightly thickened, about 3 minutes. Remove from the heat and stir in the chopped thyme and 2 tablespoons of chilled butter, 1 tablespoon at a time. Season with salt and pepper.
Bring a large saucepan of water to a boil. Add the green peas, cover and cook over moderate heat, stirring occasionally, until tender, about 3 minutes. Drain the peas, transfer half to a food processor and coarsely puree. Stir the puree back into the remaining peas. Add the reserved fat from the skillet along with the remaining 2 tablespoons of wine and the mint. Season with salt and pepper.
Transfer the chicken breasts and potatoes to plates. Spoon the mashed peas alongside and serve with the pan sauce.