Melt butter in a large skillet over medium heat; add onion, and sauté 3 minutes or until tender. Add mushrooms; sauté 2 minutes. Add cabbage and chestnuts; sauté 2 minutes. Stir in sugar and salt; cover and simmer 10 minutes or until cabbage is tender. Stir in vinegar.
*Look for whole chestnuts sold in jars at Williams-Sonoma or other cook stores.