Soak skewers in warm water at least 20 minutes while chicken is marinating.
In a large, shallow dish, mix together the yogurt, lemon juice, garlic, sugar, salt, turmeric, and pepper. Slice each chicken breast half lengthwise into 3 to 4 long strips. Add chicken to yogurt mixture and toss to coat. Cover with plastic wrap and refrigerate at least 20 minutes or overnight, if desired.
Lightly coat clean grids of grill with cooking oil spray. Heat grill to medium-high heat. Peel mangoes and slice large pieces from pits; discard pits. Remove chicken from marinade, shaking off excess, and thread on skewers. Arrange chicken diagonally across grids, without touching. Cook, with lid down, 3 minutes per side, until chicken is seared and just cooked through. Transfer chicken to a platter and cover with foil to keep warm. Place mango slices on grids and cook on both sides until just seared and warm, about 2 minutes per side. Serve skewers with grilled mango pieces.