Step 1: Activate the Yeast - In a mixing bowl, combine the lukewarm whole milk and granulated sugar. Sprinkle the active dry yeast over the top and let it sit for about 5-10 minutes until it becomes frothy.
Step 2: Mix the Dough - Once the yeast is activated, add the beaten eggs and melted butter to the mixture. Stir until well combined. Gradually add in the flour and salt, mixing until a dough forms. You may need to use your hands to knead the dough briefly until it is smooth and elastic.
Step 3: Let It Rise - Cover the bowl with a clean kitchen towel and place it in a warm area. Allow the dough to rise for about 1 to 1.5 hours, or until it has doubled in size.
Step 4: Shape the Doughnuts - Once risen, punch down the dough and turn it out onto a floured surface. Roll it out to about 1/2 inch thickness. Use a doughnut cutter to cut out shapes, placing them on a floured baking sheet as you go.
Step 5: Second Rise - Cover the shaped doughnuts with a towel and let them rise for another 30-45 minutes until they puff up.
Step 6: Heat the Oil - In a large pot or deep fryer, heat the vegetable oil to 350°F (175°C). Use a frying thermometer to ensure the temperature is just right for frying.
Step 7: Fry the Doughnuts - Carefully place a few doughnuts into the hot oil, being cautious not to overcrowd the pot. Fry them for about 1-2 minutes on each side until they are golden brown. Use a slotted spoon to remove them and place them on a cooling rack lined with paper towels.
Step 8: Prepare the Glazes - For the chocolate glaze, whisk together 1 cup powdered sugar, 2 tablespoons cocoa powder, 1 tablespoon moonshine, and enough water to reach your desired consistency. For the vanilla glaze, mix 1 cup powdered sugar, 1 tablespoon moonshine, 1/2 teaspoon vanilla extract, and water as needed.
Step 9: Glaze the Doughnuts - Once the doughnuts have cooled slightly, dip the tops into the glazes, allowing the excess to drip off. Place them back on the cooling rack to set.