In a small bowl, beat egg with a fork until combined; set aside.
Place a wire rack on a rimmed baking sheet; set aside.
Combine sugar and cinnamon in small bowl; set aside.
Lightly flour your work surface, and unroll one sheet of puff pastry dough. Brush dough with half of the beaten egg, and fold puff pastry dough up into thirds (just how it came out of the box). While dough is folded in thirds, cut out 3 donuts using a 3-inch donut cutter (or 3-inch round cutter, and then use a smaller round cutter to cut out the hole). Repeat with another sheet of puff pastry dough. Place cut out rings and holes on the prepared wire rack, and freeze for 15 minutes.
While dough is in freezer, heat oil to 375 degrees F in a large, deep saucepan over medium-high heat. When the oil is up to temperature, fry the dough 2 at a time until golden brown and puffed—fry 3-5 minutes on one side, flip over and fry for another 1-2 minutes. Drain briefly on a paper towel-lined plate, and then roll them in cinnamon sugar mixture. Repeat with remaining dough and donut holes.
MAKE BUTTERMILK GLAZE
In a medium bowl whisk together powdered sugar, buttermilk, salt, and vanilla until smooth. Thin with additional buttermilk as needed, one-half teaspoon at a time. Dip each sugared cronut into glaze and place on wire rack until ready to serve.