In the bowl of an electric stand mixer, add 1 cup of warm milk and sprinkle with the ½ Tablespoon of yeast. Let sit uncovered for 7 minutes at room temperature. Add ½ cup flour, 2 Tablespoons of sugar and whisk until blended.
Cover with plastic wrap and let rise at room temperature for 35-45 minutes (or in a 100° oven for 25 minutes). It will look puffy.
Whisk in 1 egg, remaining 2 Tablespoons of sugar, 1 Tablespoon of butter and ½ teaspoon of salt.
Using the dough hook on a stand mixer on low speed, add the remaining 2 ½ cups of flour (½ cup at a time). Let the dough blend after each addition. Add more flour 1 Tablespoon at a time, just until the dough is no longer sticking to fingertips or the walls of the bowl as it mixes for 10 minutes.
Cover bowl with plastic wrap and let the dough rise at room temperature about 2 hours (or in a 100° oven for 1 hour). Dough should be double in size.
On a well-floured surface, gently press out dough to ⅜-inch thick. Cut out dough using a 2 ½-inch donut cutter.
Set donuts on a floured baking sheet, cover lightly with a tea towel and let rise for 30 minutes.
Preheat the oil in a deep fryer or dutch oven to 350°. Gently place donuts in the oil. Fry until golden brown on both sides.
Transfer donuts to a cooling rack or paper towels placed on a baking pan. Allow to cool for 15 minutes before glazing.
