Cut X in flat side of fresh chestnuts. Place on baking sheet.
Heat oven to 400 degrees F and roast chestnuts 15 minutes or until skins begin to curl. Remove to platter; cover with towel. When cool, peel.
Quarter chestnuts. Cook in broth in saucepan, uncovered, 8 to 10 minutes, until tender. Add salt and pepper; reserve, covered.
Blanch brussels sprouts in lightly salted boiling water for 5 minutes. Drain; rinse with cold water. Slice.
Heat large skillet. Add sprouts, then butter; do not stir. Cook over high heat until sprouts caramelize, about 6 minutes; as color develops around edges, stir gently. Add chestnuts and broth from saucepan to sprouts. Serve.