01 - Dissolve yeast and 1 tablespoon sugar in warm milk in a small bowl. Let stand for 5 minutes until foamy.
02 - Combine flour, remaining sugar, salt, and nutmeg in a large bowl. Add yeast mixture, butter, eggs, and vanilla. Mix until soft dough forms.
03 - Knead dough on floured surface for 5-7 minutes until smooth and elastic. Place in greased bowl, cover, and let rise in warm place until doubled, about 1 hour.
04 - Roll dough to 1/2-inch thickness. Cut 3-inch rounds using donut cutter or two round cutters. Re-roll scraps as needed.
05 - Place donuts and holes on parchment-lined tray. Cover and let rise for 30 minutes.
06 - Mix melted butter, brown sugar, and cinnamon. Brush tops of each donut with mixture.
07 - Heat 2 inches vegetable oil in deep pot to 350°F. Fry donuts in batches, 1-2 minutes per side, until golden. Drain on paper towels.
08 - Whisk powdered sugar, milk, and vanilla until smooth. Dip cooled donuts into glaze, letting excess drip off.
09 - Immediately sprinkle donuts with purple, green, and gold sugars in sections to mimic King Cake colors. Let set before serving.