Thanks to kind reader Diana Gilman Maurer, I learned that Golden Door was indeed showing in my area, so I went to see it. It's the kind of movie that probably couldn't be made in America due to its deliberate pace, but I enjoyed it as the whole two hours follows an Italian family dealing with emigration and all it entails. You follow them as they make their decision to emigrate on through their arrival and processing at Ellis Island. With 6 of my 8 great-grandparents being Ellis Island-era arrivals, it gave me a chance to journey with them in a sense -- and I suspect fellow genealogists would appreciate this opportunity as well.
The movie also features remarkable imagery (click on the gallery here to get a sampling) -- and that's where the reference to carrots and milk comes from. But I don't want to ruin it for you, so you'll have to go watch it yourself to see what I mean!