In a deep fryer or large, heavy-bottomed pot, heat about 2 inches of vegetable oil to 350°F.
While the oil is heating, open the can of biscuit dough and separate the biscuits. Cut each biscuit into quarters, rolling each quarter into a ball to form doughnut holes.
In a small bowl, mix together the granulated sugar and cinnamon. Set aside.
Once the oil is hot, carefully drop a few doughnut holes into the oil, being careful not to overcrowd the pot. Fry the doughnut holes in batches, turning them occasionally with a slotted spoon, until they are golden brown and cooked through, about 2-3 minutes.
Using a slotted spoon, remove the doughnut holes from the oil and place them on a paper towel-lined plate to drain any excess oil.
While the doughnut holes are still warm, roll them in the cinnamon-sugar mixture until fully coated.
To make the glaze, whisk together the powdered sugar, milk (or water), and vanilla extract (if using) in a small bowl until smooth.
Dip the tops of each doughnut hole into the glaze, allowing the excess to drip off. Place the glazed doughnut holes on a wire rack to set.
Serve immediately and enjoy!