If using wood skewers, soak in cold water at least 30 minutes before using. In a large bowl or resealable plastic bag, combine chicken, cilantro, 3 tbsp. olive oil, lime juice, 2 tsp. garlic, salt, smoked paprika, and pepper. Toss to coat, then marinate, chilled, 25 minutes.
Meanwhile, put roasted peppers, nuts, bread, vinegar, remaining 1 tbsp. minced garlic, and 1/4 cup olive oil in food processor and whirl until puréed; sauce will be thick.
Preheat a gas or charcoal grill to medium-high (you can hold your hand 1 to 2 in. above cooking grate only 3 to 4 seconds). Thread chicken onto skewers, discarding marinade. Drizzle green onions with remaining 1 tbsp. olive oil. Lay skewers on grill (cover if using gas) and cook 4 minutes. Turn skewers over, then lay green onions on grill. Cook 4 minutes, until chicken is browned and cooked through and onions are charred in places. Serve hot, accompanied by sauce.
Note: Nutritional analysis is per serving.