Step 3: Cook the Carrot. Bring water to a boil in a medium saucepan. Reduce heat (until bubbles are soft) and cook carrot until tender (about 10 to 15 minutes). Drain carrot and rinse with cold water for three minutes to stop the cooking process.arrot is a very versatile veggie; after baby has mastered single-vegetable purees, you can mix carrots with a variety of fruits, other veggies, and meats. Look for firm carrots with a smooth exterior and a rich orange color. One medium carrot yields about three ounces of puree.