Add yeast and 1 tablespoon of sugar to warm milk. Mix and let sit for 5 to 10 minutes or until foamy.

In the bowl of a stand mixer, add flour, remaining sugar, and salt and mix for about 30 seconds.

Add foamy yeast milk, eggs, and vanilla to the dry ingredients and mix on low for 3 to 4 minutes or until fully combined.

Switch to the dough hook. With mixer running on low, add softened butter, one cube at a time, until all added. Wait for one cube to completely mix in before adding the next. Increase mixer to medium and continue to mix for 4 to 5 minutes, or until the dough is soft and smooth.

Place dough in an oiled bowl, cover with plastic, and place in the refrigerator overnight, or for 3 to 4 hours on the counter.

Cut two sheets of parchment paper into six pieces each and place on two sheet pans, six to a pan.

Turn the dough out onto a lightly floured surface. Gently roll it out to about a 12 inch circle, about ¼ to ½ inch thick.

Using a 3 to 3 ½ inch round cutter, cut out donuts, cutting each one as close to the previous cut as you can. Use a smaller, 1 inch, round cutter punch out the donut holes.

Gather dough scraps and roll into a ball and allow to rest for 15 minutes, to relax the gluten strands. Roll dough out and cut out as many donuts as you can. I got a total of 10 donuts, plus the donut holes. Place one cut donut on each piece of parchment and place donut holes on the other pieces.

Cover loosely with greased plastic wrap and let the donuts rise for 30 to 45 minutes.

Heat oil in a heavy Dutch oven or deep skillet to 350°F. Using the parchment paper, carefully place the donuts in the oil, to fry two at a time. The parchment paper will release and you can discard it. Cook for about 90 seconds, or until golden brown.