Cut carrots crosswise into 1/2- inch slices. Combine water and 2 teaspoons salt in a small Dutch oven; bring to a boil. Add carrots; reduce heat, and cook 5 minutes. Drain.
Add butter, onion, garlic, and remaining salt to carrots in Dutch oven. Cover; cook over low heat, stirring occasionally, 10 minutes. Remove garlic; discard. Add vinegar; mix well. Transfer to a serving bowl; sprinkle with chives. Serve hot.