Add enough oil to a Dutch oven or other frying vessel of choice to be 2 inches deep. Heat the oil to 350 degrees F (it will take a few minutes for the oil to heat up).
Put the apple juice concentrate in a small saucepan and bring to a boil over high heat. Then reduce to a low boil and cook until reduced by half and it's thickened and glaze-like, about 15 minutes. Transfer to a small heatproof bowl.
Add the sugar and cinnamon to a small bowl and stir to combine.
Remove the biscuits from the can and use a 1 1/2-inch round cutter to cut holes from the center of the biscuit dough.
Once the oil is hot, add the doughnuts 3 to 4 at a time (the doughnuts should have room to bop around). Cook until golden brown on both sides, about 2 1/2 minutes total (flipping once). Repeat with the remaining doughnuts.
Transfer to a paper-towel-lined plate to soak up the grease.
Dip both sides of the doughnut in the glaze, then sprinkle the cinnamon-sugar on the "right-side-up" side. Transfer to a cooling rack so any excess glaze can drip off. Serve warm.