In a medium bowl, add the flour, sugar, yeast, and salt. Whisk to combine, then set aside.
Heat the milk in a small saucepan over medium heat until small bubbles form around the edge. Do not boil. Add the hot milk to the bowl of a stand mixer and cool to 115℉ - 120℉.
Add the eggs to the warm milk, then add the dry ingredients. Using the dough hook attachment, mix on a low speed until a ball of dough forms, about 3 minutes.
With the mixer running on low, add about a tablespoon of butter at a time. Allow each addition to combine with the dough before adding the next. If the butter is not incorporating, stop the mixer and scrape the dough and butter down, then continue.
After all the butter has been added, increase the speed to medium and knead for 3 more minutes. If the dough is too sticky, you can add more flour a tablespoon at a time. Try to avoid adding too much flour.
The dough should form a soft ball. Add the dough to a lightly oiled bowl, cover, and allow to rise in a warm location until doubled, about 2- 2 ½ hours.
Roll the dough out to ½" thickness. Use a 3.25" circle cutter to cut out the donut rounds, then a 1.25" circle cutter to cut out the centers. Gently transfer the donuts and donut holes to a lightly floured tray and cover loosely with lightly greased clingwrap.
Gently press the dough together to form a ball. Cover and rest 10 minutes before rolling the dough back out and cutting out more donuts. Allow the donuts to rest and rise for 30- 45 minutes.
Meanwhile, add the peanut oil to a Dutch oven and bring to 350℉ - 355℉. Maintain the temperature until the donuts are done proofing.
When the donuts and oil are ready, gently transfer 1- 2 donuts into the hot oil and fry 30- 45 seconds per side. I suggest starting with one donut to determine the exact oil temperature and fry time for you. Transfer finished donuts to a paper towel lined cooling rack.
Ensure that the oil temperature is back to the right range before adding another round of donuts. Continue with this routine until all the donuts are cooked. Fry donut holes for 30 seconds per side.
Allow the donuts to cool, then proceed with making the Nutella glaze.