Place green beans in a medium saucepan; add enough water to cover green beans. Bring to boiling; reduce heat. Simmer, covered, 10 to 12 minutes until beans are very tender. Drain, reserving cooking water. Cool slightly.
Puree beans and enough reserved cooking water (1/3 to 1/2 cup) through a food mill, baby food grinder, blender, or food processor to obtain desired consistency. 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.