Light a grill or heat a grill pan. Set the turkey skin side down on a work surface. With the knife parallel to the work surface, cut through the breast, leaving 1 inch of meat attached at one side, then open it up like a book; the goal is to have meat that is of even thickness. Season with salt and pepper. Grill the turkey over moderate heat, turning once, until the skin is crisp and the meat is just cooked through, about 25 minutes.
Meanwhile, microwave 1 cup of the mango chutney until it melts slightly, about 1 minute; scrape into a food processor and puree until smooth. Transfer the pureed chutney to a bowl. In the same food processor, puree the remaining chutney with the cilantro, mint, water and oil until smooth; season with salt and pepper.
Once the turkey is just cooked through, brush it with the plain pureed chutney and grill, turning once, until the chutney forms a sticky glaze, about 2 minutes per side. Transfer the turkey to a cutting board, cover with foil and let rest for 10 minutes. Carve the turkey and serve with the cilantro-mint sauce and chutney.