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.
Like turning vegetables into candy, but surprisingly not too sweet or overly sugary. I would use whole carrots and cut into slices next time, since the baby carrots weren't quite tender enough. The chestnuts were tasty, although the sugar mixture never "dried" on them completely.
I used bottled chestnuts left over from Thanksgiving. Delicious side dish, but a little too sweet. Next time I will just skip the step of "candying" the chestnuts and add them to the carrots and glaze at the end.
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