Place beef in a medium saucepan. Add enough water to cover. Bring water to a boil. Reduce heat; simmer, covered, 2 hours or until meat is fork tender, adding carrot the last 15 to 20 minutes of cooking. Remove meat and carrot with a slotted spoon. Reserve 1/2 cup cooking liquid.
In miniature food processor, puree half the beef with 2 tablespoons reserved broth. Repeat with remaining beef and more broth. Puree carrot through a food mill, baby food grinder, or miniature food processor.
Stir beef and carrots together. Add more reserved cooking broth if desired. 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.