Warm up 1/4 cup of the milk and add the yeast and a teaspoon of sugar.Let it rest for 5-10 minutes until frothy. The milk should be hot enough to be able to place your finger in it for 5 seconds . Any hotter and your yeast will be killed. Any colder and it won't activate the yeast. If you have a food thermometer, thats between 105 to 115 F.
Stir together the flour and salt in a mixing bowl.
Meanwhile , in a saucepan heat the remaining milk, sugar and butter on low heat until the butter melts and the sugar dissolves. Take off the heat and get it lukewarm or room temperature.
Whisk in the eggs into the milk-butter mixture .
Add the activated yeast mixture and the liquid mixture to the flour.
Mix using an electric mixer fitted with dough hooks or knead by hand for few minutes until a dough is formed that pulls away from the sides of the bowl. The dough will be sticky.
Transfer to a well oiled bowl. Cover with plastic wrap or damp towel and let it rest in a warm place for about an hour or doubled in size. This is the first proofing.
When doubled in size, punch the dough to knock out the air.
Roll on a well floured surface to a thickness of about 1/4 inch. Using round cookie cutter or jar lids , cut out the doughnuts . I used 4 cm cookie cutters .
Transfer to a plate or tray and again cover and rest in a warm place for about half an hour.( second proofing)
When the dough rounds have doubled in size , heat some oil in a pan (atleast 3 inches deep) on medium or medium high. Fry the doughnuts until golden brown ( about one minute each side) .
Drain out the doughnuts on a wire rack. Let cool to room temperature.
When cool , insert a toothpick or skewer into the doughnut from one side and wiggle it inside to create some room. Fit a pastry bag with a small round tip and fill with ganache , pastry cream or jam . Insert the tip into the doughnut opening and fill until the doughnut becomes heavy or you can't push anymore ! Dust with icing sugar