Step 1: In a small bowl, combine warm milk, yeast, and 1 tablespoon of the sugar. Let sit for 5 minutes until foamy.
Step 2: In a large bowl or stand mixer, combine flour, remaining sugar, and salt. Add yeast mixture and eggs, mixing until a shaggy dough forms.
Step 3: Add softened butter one tablespoon at a time, mixing well after each addition. Knead for 8-10 minutes until smooth and elastic.
Step 4: Place dough in a greased bowl, cover with plastic wrap, and let rise in a warm place for 1-2 hours until doubled.
Step 5: Punch down dough and roll out to 1/2-inch thickness on a floured surface. Cut into 3-inch rounds using a donut cutter or two different sized round cutters.
Step 6: Place cut donuts on parchment-lined baking sheets, cover loosely, and let rise for 45-60 minutes until puffy.
Step 7: Heat 3 inches of vegetable oil in a heavy pot to 350°F. Fry donuts 2-3 at a time for 1-2 minutes per side until golden brown.
Step 8: Drain on paper towels and let cool slightly. For glaze, whisk powdered sugar, milk, and vanilla until smooth. Dip warm donuts in glaze and let set on a wire rack.