Combine first 3 ingredients in a Dutch oven; bring to a boil. Add carrots; reduce heat, and simmer, uncovered, 12 minutes or until almost tender. Drain; discard ginger slices. Set carrots aside.
Melt butter in a large skillet over medium heat. Add onion, minced ginger, and garlic; sauté 2 to 3 minutes or until onion is tender. Add reserved carrot, salt, and pepper; sauté 3 more minutes. Stir in rum, if desired, and vanilla; cook 1 minute.
Make Ahead: Place sliced carrots in a zip-top plastic bag, and store in refrigerator up to 2 days. Or for extra convenience, use packaged baby carrots.