Heat butter in a large heavy skillet over medium-high heat.
Add shallot and cook, stirring, 1 to 2 minutes, until soft and browned. Add water, syrup, salt, and cinnamon. Increase heat to high and bring to a boil, stirring occasionally. Add carrots and return to a simmer, stirring occasionally.
Cover, reduce heat to medium to maintain a lively simmer, and cook, stirring once or twice, 4 to 6 minutes, until carrots are crisp-tender.
Remove lid and stir in vinegar. Increase heat to medium-high and continue cooking, stirring often, 3 to 5 minutes, until liquid has thickened and carrots are coated.
