01 - Whisk together flour, granulated sugar, yeast, and salt in a large bowl until thoroughly blended.
02 - In a separate bowl, combine lukewarm milk, melted butter, and egg. Whisk until smooth and fully incorporated.
03 - Pour wet ingredients into dry ingredients. Mix until a soft, slightly sticky dough forms and all flour is absorbed.
04 - Turn dough onto a lightly floured surface. Knead for 5 minutes until smooth, elastic, and no longer sticky.
05 - Place dough in a lightly greased bowl. Cover with plastic wrap or a damp cloth. Let rise in a warm, draft-free area for 1 hour until doubled in size.
06 - Roll dough to ½-inch thickness. Cut into rounds using a 3-inch donut cutter and a 1-inch cutter for holes.
07 - Arrange cut donuts and holes on a parchment-lined baking sheet. Cover and let rise for 20-30 minutes until puffy.
08 - Heat vegetable oil in a heavy-bottomed pot to 350°F. Maintain temperature throughout frying.
09 - Fry donuts in batches, 1-2 minutes per side, until golden brown. Remove with a slotted spoon and drain on paper towels.
10 - Whisk powdered sugar, milk, and vanilla extract in a bowl until smooth and creamy.
11 - Dip warm donuts into glaze, allowing excess to drip off. Place on a wire rack.
12 - Immediately sprinkle sections with purple, green, and yellow colored sugars while glaze is wet. Let set completely before serving.