Open chicken and place on a work surface, skin side up. Using your palms, firmly press on breastbone to flatten the breast. Season chicken all over with 1 tablespoons salt, 1/2 teaspoons black pepper, and cayenne. Place in a baking dish; rub with 1 tablespoons oil, sprinkle with lemon zest, and scatter rosemary over. Let stand at room temperature for 1 hour.
Build a medium-low fire in a charcoal grill, or heat a gas grill to medium. Place chicken, skin side down, on grill and place a foil wrapped brick or heavy skillet on top of chicken to weigh it down. (This will expose more skin to direct heat, making it crispy; the chicken will also cook faster.) Cook until skin is crispy and golden brown, about 15 minutes.
Using tongs, set brick aside. Turn chicken, cover grill, and cook for 10 more minutes. Continue cooking and turning chicken every 10 minutes, covering grill between turns, until an instant-read thermometer inserted into the thickest part of the thigh registers 165 degrees F, about 50 minutes total. Transfer to a carving board; let chicken rest for 10–20 minutes. (Resting will make for juicier meat.) Transfer to a carving board; let chicken rest for 10 minutes before carving.
While chicken rests, add more coals to fire if necessary to increase heat to medium, or heat gas grill to medium-high. Season chile, bell pepper, and onion with salt and pepper. Grill (or use a grill basket) over medium heat, turning occasionally, until softened and charred in spots, about 12 minutes. Transfer onion to a medium bowl. Transfer chiles and peppers to a work surface and cut lengthwise into 1-inch slices. Add to bowl with onion. Add 1/4 cup vinegar to vegetables and toss to coat.
Heat 2 tablespoons oil in a small skillet over medium-low heat. Add garlic and cook until soft but not browned, about 4 minutes. Add basil and stir to wilt. Add basil mixture to bowl with onion. Season vegetables to taste with salt and more vinegar, if desired.
Rub cut side of avocados with remaining 1 tablespoons oil, season with salt, and sprinkle with chili powder. Grill, cut side down, until avocado is gently warmed and flesh is golden brown, about 3 minutes. Serve chicken with vegetables and avocados.