01 - Dissolve yeast in warm milk with a pinch of sugar in a large bowl. Let stand for 5 minutes until foamy.
02 - Add remaining sugar, egg, melted butter, salt, flour, cinnamon, and nutmeg. Mix until a soft dough forms.
03 - Knead the dough on a floured surface for 5 to 7 minutes until smooth and elastic. Place in a greased bowl, cover, and let rise in a warm place for 1 hour or until doubled in size.
04 - Preheat oven to 375°F. Line a baking sheet with parchment paper.
05 - Roll out dough to 1/2-inch thickness. Cut out donuts using a 3-inch round cutter and a 1-inch cutter for the centers.
06 - Transfer donuts to prepared baking sheet, cover, and let rise for 20 minutes.
07 - Mix brown sugar, melted butter, and cinnamon. Brush the mixture onto the tops of the donuts.
08 - Bake for 12 to 15 minutes until lightly golden. Cool on a wire rack.
09 - Whisk together powdered sugar, milk, and vanilla extract until smooth.
10 - Dip cooled donuts in icing, then immediately sprinkle with purple, green, and gold sugar in sections to mimic traditional King Cake colors.
11 - Allow icing to set completely before serving.