Brown the fennel in the rest of the oil over medium heat, turning every so often, about 15 minutes. Add the diced vegetables back to the pan, tear the basil leaves over all, add a tablespoon of the fennel greens, and pour in 1 cup water. Cover and cook until the liquid has evaporated, 10 to 12 minutes. When the pan is dry, add another 1/2 cup water and continue cooking until the fennel is tender when pierced with a knife, 15 to 20 minutes. Leave a little liquid in the pan.