Today I contacted Breyers, my go-to safe ice cream, regarding the only flavor we allow at our house, Natural Vanilla & Gluten Free flavor.  I called 1-800-931-2826 and spoke with Peter to confirm there was no cross contact risk.

To summarize, Breyers has strict processes in place to keep common allergens away from non-allergen flavors. There are dedicated, separate lines. If the manufacturing lines are shared at any time, a stringent cleaning process (with sampling) will happen before any new products are created. It will be noted on the carton with “may contain” if the lines are ever shared in the flavor you choose.
If there is no “may contain” statement, then no cross contact is happening on that line of ice cream.
It is true that multiple flavors of ice cream are all made in the same facility. They are on different lines, however.
Gluten Free flavors under-go a specialty cleaning process to certify gluten free status. There is segregation in the facility so these flavors are made on different days (which I LOVE because technically the gluten free flavors, which are all nut free, have the entire building to themselves), with an even deeper cleaning procedure. No information is available about whether or not this is a separate building or shared, or if they can 100% confirm that absolutely no other flavors are being made on another side of the facility. If I could tour this facility, I WOULD! Actually, I’d love that. Since my Dad works in food production, I can sort of picture how this would look, and I’m OK with it.
I hope this helps. Breyers has been the most cooperative and open about sharing this type of information. I’m obsessed with the cleaning process and that is why we only buy the gluten free vanilla. There are other nut-free ice creams for sale, and they are available on the SPOKIN website for you to find where they are.