Prepare the Dough:

1. In a small bowl, dissolve the yeast and 1 teaspoon of sugar in warm water. Let sit for 5-10 minutes until frothy.

2. In a large mixing bowl, whisk together warm milk, melted butter, remaining sugar, egg, egg yolk, salt, and vanilla. Add the activated yeast mixture and stir.

3. Gradually add flour, mixing until a sticky dough forms. Knead for about 8 minutes until the dough is smooth and elastic. (If using a stand mixer, use the dough hook on medium speed for 5 minutes.)

4. Place the dough in a greased bowl, cover, and let rise in a warm place for about **1-1.5 hours**, or until doubled in size.

Shape the Doughnuts:

5. Roll out the dough on a floured surface to ½-inch thickness. Use a doughnut cutter (or a large and small round cutter) to cut out doughnut shapes.

6. Place the doughnuts on a parchment-lined baking sheet, cover, and let rise for another 30-45 minutes until puffy.

Fry the Doughnuts:

7. Heat about 2 inches of vegetable oil in a heavy-bottomed pot to **350°F (175°C)**.

8. Carefully place doughnuts in the oil, frying **1-2 minutes per side**, until golden brown. Remove and drain on a paper towel-lined rack.

Make the Glaze:

9. In a bowl, whisk together powdered sugar, milk, and vanilla until smooth.

10. While doughnuts are still warm, dip each one into the glaze, letting excess drip off. Place on a rack to set for a few minutes.

Serve and Enjoy!

These doughnuts are best enjoyed fresh but can be stored in an airtight container for up to 24 hours.