In a medium bowl mix the paste ingredients. Add the chicken breasts to the bowl, turn to coat them evenly, and then transfer them to a plate, leaving any remaining paste in the bowl. Allow the chicken breasts to marinate at room temperature for 20 to 30 minutes before grilling.
Prepare the grill for direct and indirect cooking over medium heat (350° to 450°F).
Brush the cooking grates clean. Grill the chicken breasts, smooth (skin) side down first, over direct medium heat, with the lid closed, until the meat is firm to the touch and opaque all the way to the center, 8 to 12 minutes, turning once or twice. At the same time, grill the poblano and bell peppers over direct medium heat until tender, 8 to 12 minutes, turning occasionally. Remove from the grill and cut the chicken breasts into bite-sized pieces. Scrape away and discard the blackened skins from the peppers. Cut the peppers into bite-sized pieces.
Layer the tortilla chips, chicken pieces, peppers, cheese, and scallions on two large sheet pans lined with parchment paper. Place the sheet pans over indirect medium heat (you will probably need to cook the nachos in two batches), close the lid, and cook until the cheese is melted, 5 to 10 minutes. Remove from the grill, sprinkle with cilantro, and serve immediately with the salsa and the sour cream, if using.