Didn't know about the churros, but the ice cream places don't open until 11 AM at
Disneyland. I wonder if they would sell you ice cream at breakfast at Carnation Cafe though...hmm...
Now if you're in DCA, ice cream for breakfast is a definite possibility. Ghirardelli opens at the same time as the park, and they serve their full menu. I've seen people sitting side-by-side at the Ghirardelli in the early morning drinking coffee and eating ice cream.