Sprinkle flour on a clean flat kitchen surface. Take one puff pastry sheet and place it on the lightly floured surface.
With a pastry brush, brush the inside of the puff pastry sheet with egg and fold it into thirds (like you would a letter). Repeat the same process with the other puff pastry sheet and place both sheets into the freezer for 15 minutes to set.
Begin to heat the oil to 350 Degrees in a deep fryer. Remove the puff pastry from the freezer and cut out doughnuts on the puff pastry using a doughnut cutter about 1 1/2-2 inches in width. Keep the puff pastry sheet open. Leave as is. Save the scraps to make doughnut holes.
Place the doughnuts in the hot oil and fry for 5 minutes while flipping on each side until golden brown and puffed. Use a slotted spoon and remove from the oil and turn to a plate lined with a paper towel. Repeat the process for all doughnuts.
Whisk together the powdered sugar, milk, salt, and vanilla extract in a large bowl until smooth. Dip each finished doughnut into the glaze and place on a cooling rack until ready to serve.
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