Bring broth and mushrooms to a boil in a medium saucepan. Remove from heat; cover and let stand 20 minutes or until tender. Drain mushrooms in a colander over a bowl, reserving broth.
Heat a large Dutch oven over medium-high heat. Coat pan with cooking spray. Add onions; sauté 6 minutes or until lightly browned. Add garlic; sauté 1 minute. Spoon onion mixture into a large bowl.
Place flour in a shallow bowl or pie plate. Dredge pork in flour, shaking off excess. Heat oil in pan over medium-high heat. Add half of pork mixture; sprinkle with 1/8 teaspoon salt and 1/8 teaspoon pepper. Cook 6 minutes, browning on all sides. Add pork to onion mixture. Repeat procedure with remaining pork mixture, 1/8 teaspoon salt, and 1/8 teaspoon pepper.
Add wine to pan, scraping pan to loosen browned bits. Stir in reserved broth, pork mixture, and sage; bring to a boil. Cover, reduce heat, and simmer 40 minutes or until pork is almost tender.
Stir in carrot. Simmer, uncovered, 20 minutes. Stir in mushrooms, 1/2 teaspoon salt, 1/4 teaspoon pepper, and chestnuts; simmer 10 minutes.