In a small bowl or measuring glass, whisk together the warm milk, yeast, and 1 teaspoon sugar. Let sit until bubbly, 5 to 10 minutes.
In a medium mixing bowl, whisk together the flour and salt.
In large mixing bowl, whisk together melted butter, eggs, and 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), for about 5 minutes. Place the dough in a large, lightly oiled bowl. Cover and let rise in a warm place until doubled in size, for about 1 hour.
Line a baking sheet with parchment paper.Turn dough out onto a floured surface. Roll dough with a rolling pin to 1/4” to 1/3” thick and use donut cutter to make 12 donuts and donut holes. Gently press together scraps and re-roll to get another donut if desired.
Place donuts and donut holes on your prepared parchment lined baking sheet, with just enough space so they are not touching. Cover with a towel or parchment paper and let donuts rise again at room temperature for 45 minutes until puffed.
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, constantly stirring, for the donut holes. Transfer to the wire rack to drain.
Glaze
In a medium mixing 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. ENJOY!