For Chicken: Preheat oven to 425 degrees F. Remove 5 cloves from garlic head; halve remaining head horizontally. Season chicken cavity with salt and pepper; stuff with halved garlic head. Crush garlic cloves, sprinkle with 2 teaspoons salt, and, mash with flat side of a large knife to a coarse paste. Transfer to a small bowl and stir in 1 tablespoon vinegar, 1/2 teaspoon black pepper, and cumin. Rub garlic mixture over chicken. Whisk soy sauce and sugar together and brush over bird. Tuck wings under back, cross and tie legs with kitchen twine. Brush chicken with oil and place, breast side down on a rack set in a roasting pan.
Roast chicken until back is golden brown, 35 to 40 minutes. Remove pan from oven and turn chicken, breast side up. Remove kitchen twine and slightly open legs. Baste bird with pan drippings, return to oven, and roast until breast is golden brown and a meat thermometer inserted in thigh registers 170 degrees F, 20 to 25 minutes more. Transfer chicken to a carving board; let rest for 10 minutes before carving.
For Aji Verde: Puree cilantro, cheese, olive oil, water, jalapenos, garlic, vinegar, and salt in a blender until smooth. Carve chicken and serve drizzled with sauce.
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