Take a teaspoon of your total sugar and place in the warm milk followed by your yeast. Mix briefly and let sit for 10 minutes until the yeast becomes foamy.
In a bowl mix together salt, flour and set aside.
Beat your two eggs and add this to your flour mixture along with your remaining sugar and yeast mixture.
Mix well until the dough comes together. This should take about 2 minutes of mixing in a bowl and another 3 minutes kneading on a clean surface. You can do this entirely in a stand mixer using the dough attachment, mix and knead in the stand mixer for about 4 minutes.
Next incorporate your softened butter. If kneading by hand, add butter and knead for at least 10-15 minutes. Things will get very messy before the dough comes together, but be patient. If using a stand mixer add butter to dough and knead on low to medium speed until the dough comes together and is smooth and elastic. This will take some time (about 10 minutes) so be patient.
Form your dough into a ball and place in a lightly greased bowl, grease the surface of your dough with butter, cover the bowl with plastic wrap, then a clean kitchen towel over the bowl and let rise in a warm place for 1-2 hours until doubled in size.
Punch your dough down after it has risen and on a lightly flour surface, roll out to about ¼ inch to ½ inch thick. Use a cookie cutter or a biscuit cutter that is about 3 inches wide and cut out 12 circles. Re-knead the scraps, form into a ball and cover with a kitchen towel for 10 minutes so the dough can rest. This will make the dough easier to roll out. Cut out more circles, you should be able to cut out at least 2-3 more.
Place your cut out dough on parchment and cover with a clean kitchen towel and let rise again for 30 minutes.
Heat up your oil to about 350F and fry your doughnuts for about 2 and ½ minutes per side until golden brown and cooked. If your dough is browning too fast, reduce the heat. If your dough browns too fast, it will be undercooked and doughy on the inside.
Set the doughnuts on a rack to drain or place on paper towels. Let them cool completely before filling.