In a small bowl, mix together 1/4 cup oil, garam masala, ginger, garlic, and 1 tablespoon salt. Place chicken in a large baking dish or plastic bag, loosen the skin from the breast and legs of the chicken and, using your hands, spread a good part of the mixture underneath. Rub the remaining mixture inside the chicken cavity and over the skin. Tie the legs together with kitchen twine. Loosely cover, refrigerate, and let marinate 30 minutes or up to 12 hours.
Arrange the rack in the center of the oven and heat oven to 400 degrees F.
Remove chicken from marinade, let any excess drip off, and reserve marinade. Let chicken sit at room temperature while oven warms up, about 30 minutes.
When the oven is heated, place potatoes, squash, and onions in bottom of a roasting pan, drizzle with any remaining marinade, remaining 1/4 cup oil, remaining 1 tablespoon salt, and season well freshly ground black pepper. Toss to coat. Place chicken, breast side up, on top of vegetables and roast until skin becomes brown and vegetables are starting to get tender, about 1 hour. Brush yogurt on chicken and continue roasting until chicken is well browned, juices run clear when inner thigh is pierced with a knife, legs move easily in the joint, and a meat thermometer inserted into the thickest portion of the thigh registers160 to 165 degrees F on an instant-read thermometer, about another 40 to 50 minutes. Let chicken rest at least 10 to 15 minutes before serving.