1. Heat the oven to 450°. In a medium bowl, toss the Brussels sprouts with 1 tablespoon of the oil, 1/4 teaspoon of salt, and 1/4 of teaspoon pepper. Set aside.
2. Put the chicken pieces in a large roasting pan and toss with 1 tablespoon of oil, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Arrange the chicken pieces about 1 inch apart, skin-side up, and roast for 25 minutes. Add the Brussels sprouts and continue cooking until the chicken and sprouts are done, about 12 minutes longer. Transfer them to plate and leave to rest in a warm spot for about 5 minutes.
3. Meanwhile, in a medium saucepan, heat the remaining 2 tablespoons oil, the garlic, and the rosemary over low heat, stirring, for 3 minutes. Raise the heat to moderate and add the beans, 1/4 cup of the water, 1/4 teaspoon salt, and 1/8 teaspoon pepper. Cook, mashing the beans to a coarse puree, until hot, about 5 minutes. Stir in the parsley.
4. Pour off the fat from the roasting pan. Set the pan over moderate heat and add the remaining 1/4 cup water. Bring to a boil, scraping the bottom of the pan to dislodge any brown bits. Boil until reduced to 1/4 cup, about 4 minutes. Add any accumulated juices from the chicken and a pinch each of salt and pepper. Spoon the white-bean puree onto plates and top with the chicken, the Brussels sprouts, and then the pan juices.
Wine Recommendation: Pair this Mediterranean-inspired dish with a full-flavored red from France. Try one from the southern Rhône Valley such as a Châteauneuf-du-Pape or a Côtes-du-Rhône.