In the bowl of a stand mixer, combine the warm milk and yeast. Stir gently and let it sit for 5-10 minutes until foamy.

Add the flour, sugar, and salt to the yeast mixture. Mix with the dough hook on low speed until just combined.

Incorporate the eggs one at a time, mixing well after each addition, then add the vanilla extract. Increase the speed to medium-low and knead for 5-7 minutes until the dough is elastic.

Reduce the speed to low, add the softened butter cubes one at a time, allowing each piece to incorporate before adding the next. Knead until the dough is smooth and shiny.

Lightly grease a large bowl, transfer the dough, cover with plastic wrap or a damp tea towel, and let it rise in a warm place for 2-3 hours or refrigerate overnight.

Punch down the dough to release air, roll it to about 1/2-inch thickness, and cut out the donuts.

Transfer the cut donuts to parchment-lined baking sheets, cover loosely, and let rise for 45-60 minutes.

Heat the oil in a Dutch oven to 350°F (175°C), then gently lower 2-3 donuts at a time into the hot oil and fry for 1-2 minutes per side until golden brown.

Using a slotted spoon, remove the donuts and place them on a wire rack over paper towels to drain.

Dust the warm donuts with powdered sugar or brush with melted butter before serving.