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In a small bowl, whisk together the warm milk, yeast, and 1 teaspoon sugar. Let sit until bubbly, 5 to 10 minutes.
In a large bowl, whisk together the flour and salt. In another large bowl, whisk together melted butter, eggs, and remaining ¼ cup sugar until smooth. Stir in yeast mixture until combined. Stir in flour mixture until a shaggy dough forms. On a lightly floured surface, knead the dough until smooth and elastic (dough may be slightly tacky), about 5 minutes. (you can also do this in a stand mixer with the dough hook.) Place the dough in a large, lightly oiled bowl. Cover and let rise in a warm place until doubled in size, about 1 hour.
Line a large, rimmed baking sheet with parchment paper.
Punch down the dough. On a lightly floured surface, roll the dough into a ¾-inch-thick rectangle. Using a donut cutter or 3-inch and 1-inch round cookie cutters, cut 12 rounds and 12 holes in each round, re-rolling scraps as necessary. Place the cutouts on the prepared baking sheet as you work. Loosely cover and let rise in a warm place until doubled in size, about 40 minutes.
While the donuts are rising, place a wire rack inside a rimmed baking sheet. Fill a Dutch oven with 2-inches of oil. Heat over medium heat until a deep-fry thermometer registers 350F.
Working a few at a time, fry donuts in the hot oil until golden brown, about 1 minute per side for large donuts and 1 minute, constantly stirring, for the donut holes. Transfer to the wire rack to drain.
In a medium bowl, whisk together powdered sugar, milk, and vanilla until smooth. Dip warm donuts and donut holes into the glaze and return to the write rack to set.