Activate yeast mixture:
Combine the yeast, 1/2 tsp sugar, and warm water by whisking. Let stand for 5 minutes until the yeast dissolves and the liquid becomes frothy.Mix dry ingredients:
In a stand mixer bowl, whisk together the flour and salt.Combine wet and dry:
Whisk the egg yolk, milk, and vanilla into the yeast mixture, then add to the flour mixture. Use a fork to mix into a shaggy dough.Knead dough with butter:
Attach the dough hook and knead for 5 minutes, slowly incorporating the butter and scraping down the bowl as needed. The dough will be sticky and wet.First rise for 2 hours:
Cover the bowl with plastic wrap and set in a warm spot for 2 hours until the dough doubles in size.Roll dough to 1/4 inch:
Turn the dough out onto a heavily floured surface and punch down to release air. Using a well-floured rolling pin, roll to 1/4-inch thickness.Cut doughnut shapes:
Cut out doughnut shapes using the largest and smallest cookie cutters. Re-roll scraps and cut additional doughnuts.Second rest for 30 minutes:
Place the doughnuts on a floured tray, cover with plastic wrap, and let rest for 30 minutes.Heat oil and mix coating:
Meanwhile, heat at least 4 inches of oil in a large heavy pot to 360°F (180°C). In a shallow bowl, stir together the 1/2 cup sugar and cinnamon.Fry and coat doughnuts:
Fry the doughnuts in batches using a slotted spoon, cooking for 2-3 minutes per side until deep golden. Drain on paper towels, then roll in the cinnamon sugar about 30 seconds later. If rolled too soon after frying, the coating will be uneven; if too late, the sugar won't adhere.