1. Make the Dough:

1. In a small bowl, combine the warm milk, yeast, and 1 tablespoon of sugar. Let sit for 5–10 minutes until frothy.
2. In a large mixing bowl, combine the flour, remaining sugar, and salt.
3. Add the yeast mixture, eggs, and softened butter to the dry ingredients. Mix until a soft dough forms. Knead by hand for 8–10 minutes, or use a stand mixer with a dough hook for 5–7 minutes, until the dough is smooth and elastic.
4. Place the dough in a greased bowl, cover, and let rise in a warm place for 1–1.5 hours, or until doubled in size.

2. Shape the Doughnuts:

5. Punch down the dough and turn it out onto a lightly floured surface. Roll it to about 1/2-inch thickness.
6. Use a round cutter (about 3 inches in diameter) to cut out doughnuts. For classic doughnuts, use a smaller cutter (about 1 inch) to cut out the centers.
7. Place the doughnuts and holes on parchment paper-lined baking sheets. Cover loosely with a towel and let rise for 30–45 minutes, until puffy.

3. Fry the Doughnuts:

8. Heat vegetable oil in a deep pot or fryer to 350°F (175°C). Use a thermometer to maintain the temperature.
9. Carefully fry 2–3 doughnuts at a time, cooking for about 1–2 minutes per side, until golden brown. Remove with a slotted spoon and drain on paper towels. Repeat with the doughnut holes.

4. Glaze or Fill:

10. For the glaze, whisk together powdered sugar, milk, and vanilla until smooth. Dip the tops of the warm doughnuts into the glaze, then place on a wire rack to set.
11. For sugared doughnuts, roll warm doughnuts in granulated or cinnamon sugar.
12. For filled doughnuts, use a piping bag fitted with a narrow tip to inject jam, custard, or Nutella into the centers.