It’s fun to use a grill sheet (those flat metal pans with holes or slits in them) when grilling vegetables. They don’t fall through the wider grill grate during cooking. If you don’t have one, you can make your own with a piece of foil with holes cut in it. When slicing the onions, cut the wedges through the root end, so the wedges hold together.
Heat grill to medium-high heat. Mix paprika, cinnamon, and coriander in small bowl; rub over the chicken pieces. Brush vegetables with 1 tablespoon oil; season with salt and pepper.
Place chicken on grill; cover. Grill about 40 minutes, turning once, until meat reaches an internal temperature of 165°F near the bone and juices run clear. Place onions and beets on grill after turning chicken. Grill veggies until chicken is done, turning occasionally, until tender and charred.
Meanwhile, whisk 3 tablespoons oil and lemon juice together. When chicken is done, toss dressing with greens. Divide between 4 plates, top with grilled chicken and veggies. Garnish with cheese.
Nutritional values may change if you choose to substitute another chicken product. If substituting frozen chicken for this recipe, thaw before cooking.