1. If using frozen chestnuts, bring a medium saucepan of water to a boil. Add chestnuts, and boil for 3 minutes. Drain, and set aside.

2. Remove and discard stalks from fennel, reserving some fronds for garnish. Trim and discard tough outer layer if necessary. Quarter fennel bulbs lengthwise, and set aside. Peel shallots, leaving root ends intact.

3. In a large, wide saucepan, melt 1 tablespoon butter over medium-high heat. Add shallots and chestnuts, and saute stirring occasionally, until browned on all sides, about 5 minutes. Cover skillet, reduce heat to low, and cook, shaking pan occasionally until chestnuts are slightly tender, about 5 minutes. Remove shallots and chestnuts from pan; set aside.

4. Add remaining 3 tablespoons butter to pan, and melt over medium-high heat. Add fennel and sugar, and saute, turning, until browned on all sides, 12 to 13 minutes. If sugar begins to burn, reduce heat, and continue to cook.

5. Add shallots, chestnuts, stock, salt, and pepper to fennel; stir to combine, scraping up any crisp bits at 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, 12 to 15 minutes. Transfer to a serving dish, garnish with reserved fennel fronds, and serve immediately.