Well, about 63 years ago, my father allowed a 12 year old me to use his 12 ga double barrel to go crow hunting on my grandfather's farm. I had shot my 12 ga single shot with a butt pad before, but this was a momentus event to be allowed to use the double barrel by myself. Anxious...you bet! I crossed the cattle guard to the cornfield on a cold Feb. day in Oklahoma, broke the action and dropped in two Western Auto 12 ga. purple paper hull shells. I probably didn't go twenty steps and the urge to shoot somethig overtook me.
I cocked the hammers, shouldered the gun without a recoil pad, put a finger on each trigger, pointed it at a broken corn stalk and pulled the trigger. I didn't see the cornstalk disappear in a cloud of dust, but I did get a good look at the sky. When I pulled the front trigger, it caused me pull the back trigger too. Two triggers; one for each finger right? I got up, went to reload and saw I had fired both barrels, which promptly sat me down. At least I didn't drop the gun.
I was still within sight of the farm house, so I knew my dad had watched me. He never said a word, except that eventhough there were two triggers, you were only supposed to use one finger.
So, in answer to "Can you set off both barrels at the same time", I can definatively say....yes, at least in an external hammer gun.