There are two Honey Jacks. One is a meadery in Colorado that ages their mead in oak barrels and the other is a liqueur made by Jack Daniel's. Per their website, the new recipe blends sweet natural honey with the famous Old No. 7 formula for a smoother and sweeter taste that doesn't have the "bite" that the original Gentleman Jack does. I don't think you will mimic either if you attempt distilling the mead. Besides, distilling a gallon of precious mead down to a 750mL or less will make that beverage very expensive.
If all you want to do is add more alcohol to a mead, just use a lot of honey for your mead and add turbo yeast with some nutrients. keep adding honey until the yeast gives out. I have a feeling that will take a while. Good luck.
i am making both a regular distiller and will try freeze distilling shhh dont tell anyone lol. as for continuing to add more honey to feed the yeast eventually the alchohol content would get to high for the yeast to survive.