In a small bowl, combine lukewarm milk, sugar, and yeast. Stir gently and let it sit for about 5-10 minutes until the mixture becomes frothy.
In a large mixing bowl, beat the eggs. Add the yeast mixture, softened butter, salt, and vanilla extract. Gradually add the flour, mixing until a soft, sticky dough forms.
On a floured surface, knead the dough for about 5-7 minutes until it becomes smooth and elastic. Add more flour if needed, but be mindful not to make it too dry.
Place the dough in a greased bowl, cover it with a clean kitchen towel, and let it rise in a warm, draft-free place for about 1 hour or until it has doubled in size.
After the dough has risen, punch it down to release excess air. Roll it out on a floured surface to about 1/2-inch thickness. Use a donut cutter or a round cookie cutter to cut out donuts. Place the cut donuts on a baking sheet lined with parchment paper. Cover them with a kitchen towel and let them rise for another 30-45 minutes.
In a deep, heavy-bottomed pot, heat vegetable oil.
Carefully place the risen donuts into the hot oil, frying them in batches to avoid overcrowding. Fry for about 1-2 minutes per side until they are golden brown and puffed. Use a slotted spoon to remove them from the oil and place them on a paper towel-lined tray to drain any excess oil.
In a mixing bowl, whisk together powdered sugar, milk, and vanilla extract until you have a smooth glaze.
Once the donuts have cooled slightly, dip them into the glaze, allowing any excess to drip off.
Enjoy your donuts with a chilled drink.