Make-Ahead: The chicken can be prepared up to 8 hours ahead. The vegetables can be prepared up to 2 hours ahead. Assemble the nachos just before baking.
2. Spray a large nonstsick skillet with vegetable oil. Heat the skillet over medium-high heat. Remove the chicken breasts from the marinade and place in the skillet. Cook until the underside is nicely browned, about 3 minutes. Turn and cook to brown the other side, about 3 minutes more. Reduce the heat to medium and cook until the chicken springs back when pressed in the center, about 6 minutes. Remove from the heat and cool. Chop the chicken into bite-sized pieces. (The chicken can be prepared up to 8 hours ahead, cooled, covered, and refrigerated.)
3. Heat the oil in a medium saucepan over medium heat. Add the onion, red pepper, jalapeño, and garlic and cover. Cook, stirring often, until the onion is tender, about 6 minutes. (The vegetables can be prepared up to 2 hours ahead, kept at room temperature.) Position a rack in the upper third of the oven and preheat to 450°F. Spread the tortilla chips on a 12 to 14-inch diameter ovenproof platter or pizza pan. Top with the chicken, then the vegetables. Sprinkle with the cheese. Bake until the cheese melts, about 5 minutes