Combine oil and 1/4 cup vinegar in heavy large roasting pan. Add pork; turn to coat. Marinate pork 1 hour at room temperature or 2 hours in refrigerator. Transfer pork to large rimmed baking sheet; pat dry. Discard marinade in pan.
Preheat oven to 400°F. Spread mustard over top of pork, then sprinkle with salt and pepper. Roast pork until thermometer inserted into center registers 145°F, about 25 minutes. Remove pork from oven; let stand 10 minutes.
Meanwhile, melt butter in heavy medium saucepan over medium heat. Stir in chestnuts, garlic, sugar, and sage; cook 2 minutes. Add remaining 3 tablespoons vinegar and 3 tablespoons water; bring to boil. Reduce heat to low and simmer until mixture thickens slightly, about 3 minutes. Season glaze with salt and pepper.
Slice pork into 1/2-inch-thick rounds. Transfer to platter. Drizzle with warm glaze and serve.
*To prepare fresh chestnuts, start with heavy, glossy nuts with firm, smooth shells. Cut a small cross on one side of each shell to let steam escape as the nuts roast and to make them easier to peel afterward. Place chestnuts in a shallow roasting pan; sprinkle with water. Roast in 400°F oven for 25 to 20 minutes. Wrap hot chestnuts in towl and squeeze to crush the shells. Keep nuts wrapped for 5 minutes before removing the hard outer shell and the brown skin inside, being wary of burning your fingers. One pound of fresh chestnuts equals about two cups when shelled.
