Heat 1 1/4 cups milk in microwave in 30-second intervals until it is warm but not hot, about 90°F. Combine milk and yeast in the bowl of a stand mixer fitted with the dough hook.
Beat in eggs, butter, 1/4 cup granulated sugar, and salt; beat in 2 cups flour, and mix until combined. Beat in remaining flour on medium speed 3 to 5 minutes or until dough pulls away from sides of bowl.
Transfer dough to a large greased bowl, and cover. Let rise at room temperature until doubled in size, about 1 hour.
Turn out dough onto a well-floured surface; roll to 1/2-inch thickness. Cut out donuts with 2 round cookie cutters (1 large and 1 small) or a drinking glass and a shot glass, flouring cutters as you work (be sure to save the donut holes).
Knead any scraps together, being careful not to overwork the dough, and let rest before repeating the process. Divided donuts between 2 baking sheets lined with parchment paper, cover with a kitchen towel and let rise until slightly puffed, about 45 minutes.
Meanwhile, heat oil in a large Dutch oven or stock pot over medium heat until it reaches 375°F. Line 2 wire cooling racks or plates with paper towels.
Carefully add donuts and donut holes to oil, a few at a time. Cook 45 to 60 seconds or until donuts are deeply golden before very carefully flipping to other side using heat-proof tongs; cook additional 45 seconds or until deeply golden on both sides.
Transfer donuts to paper towel-lined racks or plates, and repeat with remaining donuts, being sure to keep oil temperature at 375°F.
Whisk 1 1/2 cups powdered sugar, cocoa, 3 Tbsp milk, and vanilla in a medium-sized bowl until smooth. Dip tops of donuts into glaze; top with sprinkles.
Combine 1/2 cup granulated sugar and cinnamon in another medium-sized bowl. Place donuts in bowl, turning to coat.
Add 1 1/2 cups powdered sugar to another separate bowl and place donuts in bowl, turning to coat.