01 - Whisk together flour, sugar, yeast, salt, cinnamon, and nutmeg in a large mixing bowl until thoroughly combined.
02 - In a separate bowl, blend warm milk, melted butter, and egg. Pour wet mixture into dry ingredients and mix until soft dough forms.
03 - Turn dough onto lightly floured surface and knead for 5 minutes until smooth and elastic.
04 - Place dough in lightly greased bowl, cover, and let rise in warm location for 1 hour until doubled in size.
05 - Preheat oven to 350°F and line baking sheet with parchment paper.
06 - Mix brown sugar and cinnamon in small bowl until uniform.
07 - Roll dough to 1/2-inch thickness. Cut 12 donuts using donut cutter or 3-inch and 1-inch round cutters.
08 - Arrange donuts on prepared sheet. Brush tops with softened butter and sprinkle with cinnamon-sugar mixture, pressing gently to adhere.
09 - Bake for 12-15 minutes until lightly golden. Transfer to wire rack and cool slightly.
10 - Whisk powdered sugar, milk, and vanilla until thick and smooth.
11 - Dip warm donut tops into glaze, then immediately decorate with purple, green, and gold sugars in sections. Let glaze set for 10 minutes before serving.