<<Disclaimer: Verify this information before applying it to your situation.>>

This is the reply I got from Brighams re: their ice cream....

Brigham's ice cream base does not have gluten.  The vanilla extract is
also gluten free.  Therefore the vanilla ice cream is gluten free.
As for our other flavors you should know that they are produced by the
batch and we make several different flavors in any one day.  For example
today we made:
    1. Cookie Dough ice cream
    2. Chocolate chip ice cream
    3. Oreo ice cream
As you are aware, gluten is in wheat products.  Both cookie dough and
oreo would have flour in them.  And therefore have gluten.

Since the chocolate chip ice cream was made after cookie dough, and the
chocolate chips are held in the same equipment as the cckie dough, would
the residual  effect you?  Only you can answer that question.
I do know that we always make Vanilla first and therefore it would not
be effected by any other flavors.

I hope this addresses your concerns.