Heat 2 tablespoons olive oil, 2 turns of the pan, in a large Dutch oven over medium-high heat. Season the chicken pieces with salt and pepper, and brown evenly on both sides, about 7-8 minutes total; remove to a plate. Deglaze the pan with wine and evaporate to 2-3 tablespoons; pour over chicken.
Add more olive oil to pan, 2 turns of the pan along with red onion, celery, carrots, potatoes, garlic, salt, pepper and chili pepper. Partially cover pan and stir occasionally to soften, 6-8 minutes. Add roasted peppers, tomatoes, olives, capers, vinegar, honey, basil, parsley and bring to bubble. Slide chicken and juices back in and simmer to cook through, 20 minutes at low bubble.
Serve in shallow bowls and top with toasted nuts. Pass bread at table for mopping.