01 - Combine warm water, a pinch of sugar, and yeast in a small bowl. Let stand until foamy, about 5 minutes.
02 - Whisk together milk, eggs, softened butter, sugar, salt, nutmeg, and vanilla in a mixing bowl. Add the foamy yeast mixture and stir to combine.
03 - Gradually add flour to the wet mixture, mixing until a soft dough forms. Knead by hand or with a dough hook for 8–10 minutes until smooth and elastic.
04 - Place dough in a greased bowl, cover, and let rise in a warm place until doubled in size, approximately 1–1.5 hours.
05 - Punch down dough and turn onto a lightly floured surface. Roll into a 10x20 inch rectangle.
06 - Mix brown sugar and cinnamon. Brush the dough rectangle with melted butter, then sprinkle the cinnamon-sugar mixture evenly over the surface.
07 - Roll up tightly from the long side to form a log. Pinch the seam to seal. Transfer to a parchment-lined baking sheet and shape into a ring, pinching the ends together. Optionally hide a plastic baby or bean inside from beneath.
08 - Cover and let rise until puffy, about 45 minutes. Preheat oven to 350°F.
09 - Bake for 25–30 minutes until golden brown. Cool completely on a wire rack.
10 - Stir powdered sugar, milk, and vanilla until smooth and pourable. Drizzle over the cooled cake.
11 - Immediately sprinkle purple, green, and gold sanding sugars in sections to create the traditional Mardi Gras appearance.