In a large bowl, dissolve the yeast into the warm water. Let the yeast sit and proof for about 3 minutes. Then pour in the milk and stir.
Add the sugar and egg and mix by hand.
Add 3 cups of flour and salt, and mix.
Add 3/4 cup melted (should be warm, not hot) butter and mix, then add 3 more cups of flour mixing well with wooden spoon between each one. Sometimes it works best to use clean hands for the final mixing. The dough should still be a bit sticky.
Cover the bowl with plastic wrap and let rise for one hour at room temperature.
Remove the dough from the bowl and place on a well floured surface.
Knead dough for 30 seconds, adding about 1/2 cup more flour so that it is no longer sticky.
Roll the dough out to about 1/2 inch thick on the well floured surface, checking to make sure it is not sticking as you go.
Using a 3 inch circle cutter (or the rim of a glass) and a 1 inch circle cutter, cut out the doughnuts and place them on a Silpat or parchment paper lined baking sheet. Gather the excess dough and roll out, repeating the cutting process until all the dough is used.
Cover the cut out doughnuts with a clean kitchen towel and let rise again for 30 minutes.
In a large pot, heat 2 inches of oil to 300-310 degrees F. Using a thermometer is very helpful!
Carefully place 2-3 doughnuts into the oil and fry 1-2 minutes per side.
Remove them from the oil, and let them drain on a paper towel. Dip both sides in sugar.