Activate the yeast: In a small bowl, combine the warm milk, yeast, and 1 teaspoon of sugar. Let stand for 5-10 minutes, or until foamy.

Combine dry ingredients: In a large bowl, whisk together the flour and salt.

Combine wet ingredients: In a separate bowl, beat together the remaining sugar, butter, egg, and vanilla.

Combine wet and dry ingredients: Pour the yeast mixture into the wet ingredients. Gradually add the dry ingredients to the wet ingredients, mixing until just combined.

Knead the dough: Turn the dough out onto a lightly floured surface and knead for 5-7 minutes, or until smooth and elastic.

Let the dough rise: Place the dough in a greased bowl, cover with plastic wrap, and let rise in a warm place until doubled in size, about 1 hour.

Shape the doughnuts: Punch down the dough and roll it out on a lightly floured surface to about 1/2 inch thickness. Use a doughnut cutter to cut out the doughnuts.

Let the doughnuts rise again: Place the doughnuts on a baking sheet lined with parchment paper and let rise for 15-20 minutes, or until doubled in size.

Fry the doughnuts: Heat the vegetable oil in a deep fryer or large pot to 375°F (190°C). Carefully fry the doughnuts for 1-2 minutes per side, or until golden brown.

Drain and cool: Remove the doughnuts from the oil with a slotted spoon and drain on paper towels. Let cool completely before serving.

Glazing or Topping Ideas:

Glazed doughnuts: Combine powdered sugar, milk, and vanilla to make a glaze. Drizzle over cooled doughnuts.

Cinnamon sugar: Combine cinnamon and sugar. Toss cooled doughnuts in the mixture.

Chocolate glaze: Melt chocolate chips and a little butter. Drizzle over cooled doughnuts.

Enjoy your homemade doughnuts!