In a large bowl, whisk together the soy sauce, 1 tbsp oil, cilantro, ginger, garlic, and chili paste. Add the chicken breasts and toss to coat well. Cover and refrigerate for at least 30 minutes or up to 1 day.
Preheat the grill or grill pan.
Meanwhile, cut the bell peppers into quarters. Discard the stems and seeds. Place the peppers on the greased grill over medium-high heat. Grill until the skins start to blacken, about 15 minutes. Remove to a plate. Place the chicken breasts on the greased grill over medium-high heat and grill, turning once, until no longer pink inside, 4-6 minutes per side. Remove to a plate.
Chop the grilled peppers and chicken into bite-size pieces. n a large bowl, toss the chicken and peppers with the greens, the remaining oil, the vinegar and salt.