Add the eggs and melted butter and mix them in. While the mixer is running on medium-low, stir in the flour and salt until completely combined, about five minutes.  Turn off the mixer and allow it to rest for 10 minutes.

After it has rested, move the dough to a clean, oiled bowl and turn it to coat all of the dough in oil.  Cover the bowl with plastic wrap and put in the refrigerator for at least two hours, or up to 1 day.

To make the donuts, remove the dough from the refrigerator and turn it out onto a floured surface.  With a rolling pin, roll it out until it is about 1/3-1/2″ thick, then using a large circle cutter (something about 3-4 inches, I like to use a wide-mouth jar lid), cut as many donuts as you can. Using a small cutter (about 1″), cut a hole out of the center of each donut.  Lay them out on parchment lined baking sheets, leaving about an inch between them.  Repeat the rolling & cutting with the dough edges until you have used it all.

Gently lay plastic wrap over the top of the doughnuts, and let them rise for 1-2 hours, or until they are puffed up and soft when you touch them.  If your house is cold, turn on your oven for about 2 minutes then turn it off, just to make it slightly warm.  Place the trays of donuts in the oven.

In a medium, heavy-bottomed pan, heat at least 2″ of oil to 350°.  Carefully drop 2 or 3 of the donuts in and fry for about 2 minutes on each side, or until each side is golden brown.  Remove from the oil to a cooling rack and repeat with the rest of the donuts and donut holes, keeping the oil around 350°.