Place carrot in a small saucepan. Add enough water to cover. Bring water to a boil. Reduce heat to medium and simmer, covered, 15 to 20 minutes or until carrot is very tender. Drain carrot, reserving cooking liquid.
Puree carrot with 2 tablespoons of the cooking liquid through a food mill, baby food grinder, or miniature food processor. Add additional cooking liquid as needed for the consistency your child likes. Use within 1 day, or freeze up to 1 month.
Spoon finished food into ice cube trays. Cover with waxed paper and freeze until firm. When ice cubes are frozen, remove cubes from tray and store in a resealable plastic freezer bag or freezer container. Label package with contents and date. Use within 1 month. Before serving, let thaw in the refrigerator overnight.
Heat thawed mixture to lukewarm in a small dish set in a pan of hot water. Stir before serving. Always sample a small bite to make sure the food is not too hot.