Method:
Put lime juice, 1/4 cup oil, garlic, cilantro, 1/2 teaspoon salt and chicken into a sturdy, resealable plastic bag; seal tightly, shake to coat chicken and refrigerate at least 1 hour or overnight.
Arrange an oven rack 6 inches from the broiler; preheat the broiler. Heat remaining 2 tablespoons oil in a large skillet over medium heat. Add onions, peppers and remaining 1/2 teaspoon salt and cook, stirring occasionally, until onions are soft and golden brown, about 15 minutes. Meanwhile, remove chicken from its marinade and transfer to a foil-lined baking sheet; discard leftover marinade. Broil chicken, flipping once, until just cooked through, about 5 minutes per side. Transfer to a cutting board, set aside to rest for 5 minutes, then slice thinly.
Arrange chicken on a platter with onions and peppers. Serve with tortillas, salsa and cheese on the side so that everyone can assemble their own fajitas.