Combine 2 tablespoons sugar and 2 teaspoons butter in a medium nonstick skillet over medium-high heat; bring to a boil. Add chestnuts; cook 2 minutes or until browned, stirring often. Spread the chestnuts in a single layer on wax paper, and let stand at room temperature until dry.
Heat 2 teaspoons butter and oil in a large nonstick skillet over medium-high heat. Add carrots; saute 5 minutes. Reduce heat to medium-low. Stir in 3 tablespoons sugar; cook 10 minutes, stirring frequently. Add broth; simmer 2 minutes. Stir in parsley and salt. Top with chestnuts.