This is a normal (although extremely painful) condition called Duratus Renibus. It usually occurs when someone ingests extremely cold substances i.e. Ice Cream, Slushes, cold beverages. There is currently no known medical explanation for this anomaly.
By the time the ice cream or slurpee or whatever is eaten and swallowed, the temp of the food has warmed up considerably. It's not going to be cold enough to trigger any type of vasospasm in the kidney. The cold foods right in the mouth chilling the soft palate does cause the "brain freeze" as it's commonly called. Perhaps it's something else like a sugar spike but even that takes time for the sugar content to enter the blood stream.
I too get that but on both sides, I get them eating ice cream, frozen drinks, etc. I ask my doctor about it and he thought I was crazy.