1Preheat oven to 400ºF. Set chicken pieces on a roasting pan. Brush with olive oil (about 2 tablespoons total) on both sides. Season both sides with salt (about 1 teaspoon total) and several grinds of pepper.
Put peppers, onions and carrots in a large bowl. Toss with 2 tablespoons olive oil and 1 teaspoon kosher salt. Spread them out on a roasting pan. Cook vegetables on the upper shelf of the oven and chicken on the lower shelf. After 25 minutes, remove the chicken and let it cool. Move the vegetables to the lower shelf, toss them, and roast for an additional 15 minutes. When chicken is cool enough to handle, cut it into bite-sized pieces and set aside.
Meanwhile (while everything is roasting) heat 2 tablespoons olive oil in a large dutch oven or soup kettle over medium heat. Add garlic and red pepper flakes and cook for about 30 seconds, just until the garlic is fragrant. Add mushrooms and cook, stirring often, for about 5 minutes, until the mushrooms lose their water and start to brown. Add white wine and cook, stirring, for 30 seconds or so. Stir in broth, tomatoes, tomato paste, spices and salt. Bring to a boil, lower heat and simmer for 10 minutes. Add potatoes, chicken and capers. Bring to a gentle simmer and cook, uncovered, for 35 minutes, until potatoes are tender and the sauce has thickened a bit. Season, to taste, with salt and pepper.