Line a baking sheet with paper towels and set a cooling rack on top. Add enough oil to a Dutch oven or deep-sided skillet to measure a depth of at least 3-inches. Heat the oil over medium high heat until it reaches 350ºF (170ºC) on a candy thermometer.
While the oil is heating, cut the donuts. Lightly flour a baking sheet. Roll the dough out on a well-floured surface to about 1/2 inch thick. Use a well-floured donut cutter to cut as many donuts and holes as possible; the dough will be soft and sticky, so do your best to handle it gently. Flour the cutter in between each cut to prevent sticking. Gently reroll the scraps and cut again. Transfer the cut donuts and holes to the floured baking sheet. Refrigerate while the oil heats.
When the oil has come up to temperature, dust the excess flour off the donuts and holes and carefully place each one in the hot oil, working in batches and making sure to not crowd the pan; the donuts will expand a bit in the oil. Keep an eye on the temperature of the oil; you may have to raise or lower the heat under the pan to keep the oil at a consistent temperature. Refrigerate the baking sheet with the uncooked donuts in between batches.
Fry each batch of donuts and holes until they are deep golden brown, about 90 seconds, then use a spatula or spider strainer to carefully flip the donuts. Cook the other side for about 90 seconds, or until deep golden brown and cooked through. Remove to the rack until to cool enough to touch, about 5 minutes.
Dip each warm donut in the glaze and let the excess drip off. Return the glazed donuts to the rack, glaze side up, and let them sit until the glaze is no longer sticky. Serve warm or at room temperature. These donuts are best the day they are made.