have no idea of the origin of this pastry. I believe that the French originated
the idea of deep-fried foods (i.e., the french-fry) but beyond that I don't
know whether this is French-Canadian or Acadian. I suspect that this came
later through our French-Canadian cousins. I doubt seriously that the Acadians
might have used the deep-frying technique. But, the beigne is basically
the doughnut which is deep-fried.
My mother would make these by the dozens (7-8 dozens a week)... and she barely kept up with us. We would typically have them after breakfast or as a snack.
When the doughnut looks like it is done, take one out and open the inside to see if it is cooked all the way through. If there is still some raw batter in the middle. Continue cooking.