01 - In a large mixing bowl, dissolve yeast in warm milk with a pinch of sugar. Allow to stand for 5 minutes until foamy and activated.
02 - Add remaining sugar, egg, melted butter, salt, flour, cinnamon, and nutmeg to the yeast mixture. Mix until a soft dough forms.
03 - Transfer dough to a floured work surface and knead for 5 to 7 minutes until smooth and elastic. Place in a greased bowl, cover with a damp cloth, and let rise in a warm location for 1 hour or until volume doubles.
04 - Preheat oven to 375°F. Line a baking sheet with parchment paper.
05 - Roll out dough to 1/2 inch thickness. Cut donuts using a 3-inch round cutter and a 1-inch cutter for 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 over donut tops.
08 - Bake for 12 to 15 minutes until lightly golden. Transfer to wire rack to cool completely.
09 - Whisk together powdered sugar, milk, and vanilla extract until smooth and pourable.
10 - Dip cooled donuts into icing, then immediately sprinkle purple, green, and gold sugar in sections to represent traditional King Cake colors.
11 - Allow icing to set completely before serving.