Rinse turkey, and pat dry with paper towels. Place it on a flat surface, skin side up. Using a chopstick or the handle of a wooden spoon, gently separate turkey skin from breast meat, being careful not to tear the skin. Remove skin, and transfer to a parchment-lined baking sheet.
Flip turkey over, and lay flat. If turkey is not an even thickness, cover with plastic wrap, and gently pound with a mallet or a rolling pin. Season with salt and pepper. Spread stuffing evenly on turkey. Starting on 1 short side, tightly roll up turkey.
Center reserved skin on turkey roll, and smooth to remove any trapped air. Season with salt and pepper. Wrap entire turkey roll in cheesecloth, securing both ends with kitchen string. Tie additional string around roll in 3 or 4 places, spacing evenly. Rub with 4 tablespoons butter.
Preheat oven to 375 degrees with rack in lowest position. Place turkey roll on a rack on a rimmed baking sheet. Transfer to oven, and pour water in baking sheet. Roast, rotating halfway through and brushing with remaining 2 tablespoons butter, until stuffing registers 165 degrees on an instant-read thermometer, about 2 hours and 10 minutes. (Add more water to baking sheet if necessary to prevent pan drippings from burning.) Let turkey roll rest on baking sheet for 30 minutes.
To carve the turkey roll: Remove cheesecloth, and transfer turkey roll to a cutting board. Carefully cut roll into slices at least 1/2 inch thick, keeping stuffing intact. Transfer slices to a serving platter, and serve immediately with chutney on the side.
To save time, ask your butcher to bone and butterfly the turkey breast. You may want to make this request at least one or two weeks in advance, especially during the holidays. Make-ahead tip: Prepare the stuffing the day before. Spread it on turkey, and roll, cover, tie, and refrigerate.