Heat oil in small skillet until shimmering. Add garlic and ginger and cook until fragrant, about 1 min. Add garam masala, cumin, and chili powder; continue to cook until fragrant, 30-60 sec longer. Transfer half to small bowl and stir in yogurt and 2 Tbsp lime juice. Set aside.
In bowl large enough to hold all the chicken combine remaining garlic-spice mixture, remaining 2 Tbsp lime juice, and salt. Using sharp knife, lightly score skinned side of each piece of chicken, making 2 or 3 shallow cuts about 1“ apart and about 1/8” deep; transfer to bowl. Using hands, gently massage salt-spice mixture into chicken until all pieces are evenly coated; let stand at room temperature 30 min.
Adjust oven rack to upper position, about 6“ from element, and heat oven to 325 deg. Pour yogurt mixture over chicken and toss until chicken is evenly coated with thick layer. Arrange chicken pieces, scored-side down, on wire rack set in foil-lined, rimmed baking sheet or broiler pan. Discard excess yogurt mixture.
Bake until instant read thermometer inserted into thickest part of chicken registers 125 deg for breasts or 130 deg for legs and thighs, 15-25 min. Transfer pieces to plate when they reach temperature, smaller pieces will be done sooner.
After removing chicken from oven, turn oven to broil and heat 10 min. Once broiler is hot, flip chicken pieces over and broil until chicken is lightly charred in spots and thickest part registers 165 deg for breasts and 175 deg for legs and thighs, 8-15 min. Transfer chicken to large plate, tent loosely with foil, and rest 5 min. Serve with basmati rice chutney or relish.