1. In a large bowl, whisk 1 cup of the yogurt with the cider vinegar, 2 tablespoons of the lemon juice, two-thirds of the minced garlic cloves, 1 teaspoon of salt, 1 teaspoon of black pepper, the cardamom and cayenne. Add the chicken breasts and turn to coat. Cover and refrigerate for 4 hours.
2. Meanwhile, in a small bowl, whisk the tahini paste with the remaining minced garlic and 3 tablespoons each of yogurt and lemon juice. Season with salt and pepper.
3. Light a grill or heat a grill pan. Brush the grill grates or the pan with the canola oil. Remove the chicken from the marinade and shake off any excess. Grill the chicken breasts over moderate heat, turning once, until they are browned and cooked through, about 15 minutes. Transfer the chicken breasts to a cutting board and let them rest for 5 minutes. Slice the chicken into strips and serve with the tahini sauce.