Preheat grill to medium-high heat. After preheating, reduce one side of grill to low heat.
Combine 1/4 cup green onions, 1/4 cup shallots, sugar, 2 tablespoons juice, oil, allspice, garlic, and serrano in a mini food processor; process until smooth. Combine half of onion mixture and chicken in a medium bowl; toss well. Sprinkle with salt.
Place chicken on grill rack coated with cooking spray over medium-high heat. Cover and grill 5 minutes on each side. Move chicken over low heat. Cover and grill 5 minutes on each side or until done. Brush chicken with remaining onion mixture.
Combine remaining green onions, remaining shallots, remaining juice, cheese, cilantro, and sour cream. Halve bell peppers lengthwise; discard seeds. Divide cheese mixture evenly among pepper halves. Place peppers on grill rack coated with cooking spray over medium-high heat. Grill 7 minutes or until peppers are lightly charred.