Pour about 2 inches of oil in a large, heavy-bottomed kettle (like a dutch oven) and heat over medium heat until 335°. Line a baking sheet with paper towels and place a cooling rack on top of it. Set aside.
While the oil is heating, lay out a sheet of parchment paper and place the chilled dough on it. Place another sheet of parchment on top and roll to 1/2-inch thickness.
Using a 3-inch round cookie or biscuit cutter dipped in flour, cut circles out of the dough. Cut holes in the center of each circle using a 1 1/4-inch round cutter. (The large end of a piping tip works well too!) Place the doughnuts and holes on another sheet of parchment paper.
When the oil is heated, carefully place 2-3 doughnuts in the kettle (or until it's comfortably full but you can still flip them easily). They may sink to the bottom at first, but should float in about 5-10 seconds.
Fry for 2 minutes, then flip and fry for another 2 minutes. Flip one more time and fry for another 30-60 seconds, or until golden brown.
Remove the doughnuts with a slotted spoon and place on the cooling rack.Adjust the temperature as needed to keep the oil at 335° for the remaining batches of doughnuts. Combine the glaze ingredients in a bowl and dip the doughnuts in the glaze while they are still very warm.