Heat the oven to 425°F and line an 18-by-13-inch rimmed sheet pan with foil. Toss fennel with 1/2 tablespoon oil and 1/4 teaspoon salt and spread across the center.
Pat chicken dry. Rub with remaining oil, lemon zest, garlic, thyme, pepper and remaining salt, then set skin-side up with the fennel around the pieces.
Roast for 25 minutes. Mix beans with 1 tablespoon lemon juice and the measured water, then spoon around the chicken and turn the exposed fennel wedges.
Roast for 15 to 20 minutes more until the thickest breast meat reaches 165°F and fennel is tender. Check temperature away from the bone.
Rest chicken for 5 minutes. Drizzle remaining lemon juice over the beans and fennel and serve 1 breast half with half the vegetables and beans per portion.