01 - Whisk together flour, granulated sugar, yeast, salt, cinnamon, and nutmeg in a large bowl until evenly combined.
02 - Combine warm milk, melted butter, and egg in a separate bowl. Pour 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 place 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 for filling.
07 - Roll dough to 1/2-inch thickness. Cut 12 donuts using 3-inch and 1-inch round cutters.
08 - Place 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. Cool slightly on wire rack.
10 - Whisk powdered sugar, 2-3 tablespoons milk, and vanilla until thick glaze forms.
11 - Dip warm donut tops into glaze. Immediately sprinkle with purple, green, and gold sugars in sections to create King Cake pattern. Let set 10 minutes before serving.