1. Remove and discard the stalks from the fennel, reserving some of the fronds for garnish. Trim and discard the tough outer layer, if necessary. Quarter the bulbs lengthwise, and set them aside.
2. In a large, wide saucepan, melt 1 tablespoon butter over medium-high heat. Add the shallots and chestnuts, and saute, stirring occasionally, for 5 minutes or until the chestnuts are browned on all sides. Cover the pan, reduce the heat to low, and cook, shaking the pan occasionally until the chestnuts are slightly tender, about 5 minutes. Remove the shallots and chestnuts from the pan, and set aside.
3. Add the remaining 3 tablespoons butter to the pan, and melt it over medium-high heat. Add the fennel and sugar, and saute until fennel is well browned on all sides, about 12 to 13 minutes. If the sugar begins to burn, reduce heat and continue to cook.
4. Add shallots, chestnuts, broth, salt, and pepper to the fennel; stir to combine, removing any crispy bits from the bottom of the pan. Cover and bring to a boil. Reduce heat to medium, and cook, carefully turning vegetables occasionally, until they are tender, about 20 minutes. Uncover, raise heat to medium high, and cook until liquid is reduced to a glaze, about 12 to 15 minutes. Transfer to a serving dish, garnish with reserved fennel fronds, and serve immediately.