In a small microwave-safe bowl, combine the milk and water and heat for a few seconds in the microwave to lukewarm.
Whisk in the yeast with the warm milk and water and let stand for 5 minutes until it starts to bubble.
Add the flour, sugar, salt, and 1 1/2 tablespoons softened butter to a stand mixer bowl.
Add the yeast mixture to the dry ingredients and mix with the paddle attachment until just combined, about 2 minutes.
Scrape down the sides of the bowl and mix again for another minute. Press the dough into a ball and place in a greased bowl.
Cover with a damp cloth or loosely with plastic wrap and place in a warm spot. Let rise until doubled in size, about 45 minutes. At this point, punch down the dough, wrap and chill for 1 hour.
While your waiting, place the cup of butter between 2 pieces of parchment or wax paper. Use a rolling pin to flatten and shape into an 8-by-8-inch square.
Once your butter block is ready, roll out the chilled dough on a lightly floured surface into a 12-inch by 12-inch square. Place the butter square on top of the dough square so that the corners of the butter block are pointing to the sides of the dough square.
Fold the corners of the dough over the butter block to meet in the center. Use your fingers to pinch together the dough at the seams so the butter block is completely sealed.
Roll dough out into a 20-by-8-inch rectangle. (If the dough is too stiff, smack it a few times with the rolling pin to soften.)
Brush off any excess flour and fold the dough into thirds, like a letter. Cover with plastic wrap and chill for 1 hour.
Repeat the folding and chilling process 2 more times.
Once dough is chilled after its final fold, roll it out on a lightly floured surface into a 6-by-18-inch rectangle. Use a very sharp knife or pizza cutter to cut the dough into 3 6-by-6-inch squares and stack the squares on top of each other.
Roll the dough into a 6-by-8-inch rectangle. All this folding and stacking creates layers and layers of buttery croissant dough.
Using donut ring cutters, cut 12 donut shapes out of the dough and place on a parchment-lined baking sheet. Cover the donuts loosely with plastic wrap and let rise in a warm place for about 30 minutes.
Place the oil in a large pot and heat to 350 degrees. Fry donuts for about 1½ minutes, flipping them in the oil halfway through, until they are golden brown all over. Don't crowd the pan, fry in batches if needed. Place on a paper towel lined plate to cool.