Preheat a grill or grill pan to medium. In a shallow bowl, pour all but 2 teaspoons of the lime juice over the chicken. Add 1 tablespoon olive oil and turn to coat; season with salt and black pepper. Cover and let marinate at room temperature for 15 minutes.
Meanwhile, in a large bowl, toss the chiles and onion with 2 tablespoons olive oil; season with salt and black pepper. Transfer the chiles to the grill and tent with foil. Grill, turning occasionally, until softened, 10 to 15 minutes. Transfer to a large bowl, cover and let stand for 15 minutes. Meanwhile, grill the chicken and the onion, turning once, 5 to 6 minutes on each side; let cool.
Peel the chiles and slice. Using a food processor, puree one-third of the chiles with the buttermilk, 1 tablespoon olive oil and the remaining 2 teaspoons lime juice; season with salt and black pepper. Transfer the remaining chiles to a medium bowl. Coarsely chop the grilled onion and add to the sliced chiles.
Brush the tortillas with the remaining 2 tablespoons olive oil and season with the cayenne, salt and black pepper. Grill, turning once, until crisp, about 2 minutes; cut into wedges.
In a large bowl, toss the romaine lettuce with half of the poblano dressing. Slice the chicken and serve on top of the romaine. Add the chile-onion mixture and the remaining dressing; serve with the grilled tortilla chips.