For the Dough:

1. Begin by whisking the warm milk, yeast, and 1 teaspoon of sugar in a small bowl. Allow it to sit until it becomes bubbly, which should take about 5 to 10 minutes.
2. In a large mixing bowl, combine the flour and salt. In another large bowl, blend the melted butter, eggs, and the remaining ¼ cup (50g) of sugar until the mixture is smooth. Incorporate the yeast mixture until everything is well combined. Gradually add the flour mixture until a shaggy dough forms. On a floured surface, knead the dough until it is smooth and elastic, which should take around 5 minutes. (This can also be done using a stand mixer with a dough hook.) Transfer the dough to a large, lightly oiled bowl. Cover it and let it rise in a warm area until it doubles in size, approximately 1 hour.
3. Prepare two large, rimmed baking sheets by lining them with parchment paper.
4. After the dough has risen, punch it down. On a floured surface, roll it out into a rectangle about ¾-inch thick. Use a donut cutter or a pair of round cookie cutters (3-inch and 1-inch) to cut out 12 rounds and 12 holes from each round, re-rolling scraps as needed. Place the cutouts on the prepared baking sheets as you go. Loosely cover them and let rise in a warm location until they double in size, roughly 40 minutes.
5. While the donuts are rising, set a wire rack inside a rimmed baking sheet. Fill a Dutch oven with 2 inches of oil and heat it over medium heat until a deep fry thermometer reads 350°F.
6. Fry the donuts in small batches in the hot oil until they turn golden brown, which should take about 1 minute per side for the larger donuts and about 1 minute, stirring constantly, for the donut holes. Move them to the wire rack to drain.

For the Glaze:

1. In a medium bowl, combine the confectioners’ sugar, milk, and vanilla, whisking until smooth. Dip the warm donuts and donut holes in the glaze, then return them to the wire rack to allow the glaze to set.