Mix everything really well until the yeast is dissolved.
When the yeast is dissolved, add the melted but cool butter, eggs and vanilla extract.
Using a whisk, stir everything well.
Add the flour and salt and mix well until you obtain a dough. The dough will be sticky.
Start kneading the dough for 10-15 minutes until you obtain a smooth, elastic dough that is just a little bit sticky.
If the dough is too sticky, just add a little bit of flour, if it is too dense, add a little bit of milk. However, add little by little and knead until you get the right consistency.
Keep the dough in the bowl but cover it with foil and let it rise in a warm place for about 40 minutes, or until it doubles in size.
When the dough has proofed, sprinkle some flour on the table and place the dough on it.
Split the dough in 10 pieces and roll them in your hands to make them round. Place them on the table.
Take a piece of dough and roll it into a cord. Fold the cord in half and twist it to form the doughnut.
Press the ends together and place the doughnut back on the table.
Continue the same with the rest of them.
Let them proof again on the table for 15-20 minutes, while you heat up the oil in a deep frying pan.
Place the doughnuts in the heated oil and fry them on medium heat for 3-4 minutes on each side, making sure they cook all the way around. Do not crowd them, as they will need some space to rise. When cooked, they should be golden brown.
Remove them from the pan and place them on paper towels to drain.
While still hot, sprinkle some powdered sugar on the doughnuts and serve.