Welcome to the forum!
You probably get a bout of sinusitis and inflammation of the throat with anything cold. I would suggest you make it a practice to gargle with warm saline water everyday at bedtime. Also if you take something cold, generally these problems start after 10-12 hours. Almost always, a person gets up next day to find his/her throat inflamed. Hence if you gargle after an hour of taking ice cream and before bedtime you will be able to prevent it.
However if you get the symptoms soon after eating something cold, then there is a chance that you have a very sensitive throat or a lingering infection and you should consult an ENT.
Hope this helps. Please let me know if there is any thing else and do keep me posted. Take care!