In a medium saucepan, cook chicken breast half in enough boiling water to cover for 15 minutes or until chicken is no longer pink (internal temperature should be 170°F), adding the carrot to the boiling water the last 10 minutes of cooking. Drain chicken and carrot, reserving cooking liquid. Cool chicken and carrots.
Finely chop chicken and carrots and combine in a medium bowl with the cooked pasta. Add just enough of the reserved cooking liquid to moisten. 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.