Pour the warm milk into a small bowl and sprinkle the active dry yeast over the top. Add one teaspoon of sugar from your measured amount, then gently stir. Let the mixture sit for about 8 to 10 minutes undisturbed.

In a large mixing bowl, combine the flour, remaining sugar, and salt. Whisk them together briefly so everything is evenly distributed. Then make a well in the center of the dry ingredients and pour in the activated yeast mixture.

Crack both eggs into the well and add the vanilla extract. Using a wooden spoon or a dough hook on your stand mixer, start mixing everything together. Once a rough dough forms, add the softened butter one tablespoon at a time, mixing well after each addition.

Keep mixing for about 5 to 7 minutes until the dough becomes smooth, elastic, and just slightly tacky to the touch.

Shape the dough into a smooth ball and place it in a lightly greased bowl. Cover the bowl with a clean kitchen towel or plastic wrap. Set it in a warm, draft-free spot and let it rise for about 1 hour to 1 hour 15 minutes.

Once the dough has risen, punch it down gently to release the air. Turn it out onto a lightly floured surface and use a rolling pin to roll it out to about 1/2-inch thickness.

Use a 3-inch round cutter to cut out your donuts and a 1-inch cutter for the center holes. Arrange the cut donuts and donut holes on your prepared baking sheets, spacing them at least 2 inches apart.

Cover the cut donuts loosely with a towel and let them rest for another 20 to 30 minutes.

Pour the neutral oil into a deep pot and heat until it reaches 350 to 360 degrees Fahrenheit.

Carefully slide 2 to 3 donuts into the hot oil at a time and fry each donut for about 1 minute and 30 seconds per side.

Transfer the fried donuts immediately to a paper towel-lined tray and let them drain and cool for about 3 to 4 minutes before glazing.

While the donuts are still slightly warm, whisk together the powdered sugar, milk, and vanilla extract in a shallow bowl until completely smooth.

Dip each donut face-down into the glaze, hold it for 2 to 3 seconds, then let the excess drip off.