To decorate the cake:

1 cup powdered sugar, sifted

1/4 teaspoon almond extract

2 tablespoons whole or 2% milk

Purple, green, and yellow paste food coloring (or other colors depending upon the occasion)

Purple, green, and yellow sprinkles

Equipment:

Stand mixer, or large mixing bowl if kneading by hand

Rolling pin

Spatula

Pastry scraper

Baking sheet

Parchment

Cooling rack

Small mixing bowls, for mixing the icing

Mix the dough: Combine the water and yeast in the bowl of a stand mixer and let it dissolve. Stir in the softened butter, milk, eggs, sugar, nutmeg and salt. Add 3 cups of flour and stir to form a shaggy dough. Reserve the remaining 1/2 cup of flour.

Knead the dough: Using the dough hook attachment, begin kneading the dough on low speed. Continue kneading the dough until all the loose flour has been worked in, the dough comes together in a ball, and it feels smooth and elastic, 5 to 7 minutes. It will feel a bit more dense and malleable than regular bread dough -- more like play dough. Alternatively, turn the dough out on a lightly floured counter and knead by hand for about 10 minutes. If the dough sticks to the bowl, the hook, or your hands, add a little of the reserved flour.

Let the dough rise: Clean the bowl used to mix the dough and film it lightly with vegetable oil. Shape the dough into a ball with your hands, set it in the bowl, and cover. Place the bowl somewhere warm and let the dough rise until doubled in bulk, about 1 hour.

Roll the dough into a rectangle: Once the dough has doubled in bulk, scrape it out onto a lightly floured counter. Pat it into a rough rectangular shape using floured hands, then use a rolling pin to roll the dough into a fairly thin rectangle roughly 24 inches wide by 10 inches tall.

Spread the filling over the dough: Use a spatula to spread the butter in a thin layer all over the dough. Leave 1/2 inch of clear space at the top of the rectangle. Whisk together the sugar and cinnamon, then sprinkle it evenly over the butter. Press lightly with the palms of your hands or the rolling pin to help the sugar stick to the butter.

Hide the bean: Place the dried red bean or pecan half somewhere on the surface of the dough; it will be hidden once you roll up the dough.

Roll up the dough: Starting at the long end closest to you, roll the dough up toward the top. A pastry scraper can help with this step if your dough sticks. Pinch the log of dough closed and roll it over onto the seam.

Coil the log into a circle: Curl the log of dough into a ring and overlap the ends slightly so that the ring is the same height all around. Pinch the dough lightly where the ends overlap so they form a seal. Transfer the round to a baking sheet lined with parchment.

Let the dough rise: Drape the dough loosely with plastic wrap. Let it rise at room temperature until it looks puffed and slightly larger, 45 to 60 minutes.

Preheat the oven to 350 F halfway through the dough rising.

Bake the king cake: Bake until lightly browned and golden on top, 30 to 40 minutes. A little syrupy filling might ooze onto the baking sheet; this is normal.

Cool the cake: Let the cake cool for a few minutes on the baking sheet, then transfer the cake to a cooling rack. Let it cool completely to room temperature.

Prepare the icing: Mix together the powdered sugar, almond extract and 2 tablespoons of milk in a small bowl to create a thick, but pourable, icing. Add 1/2 tablespoon more milk as needed to adjust the consistency. Leave the icing white if you're decorating the cake with colored sprinkles, or divide the icing into three bowls and tint each one a different color with food dyes.

Decorate the cake: Drizzle the frosting over the cake in patches of different color, letting some icing drizzle down the sides. "Paint" the cake with as many different patches or strips as you like. Cover with colored sprinkles, if you like.

Store or serve: This cake is best served within a day or two of making it. Let the icing dry completely, then loosely cover with plastic and store at room temperature -- avoid the plastic touching the icing as much as possible, or wait to frost until just before serving. The sliced cake or leftovers should be kept in an airtight container at room temperature.

Recipe notes: Let the cake cool completely, then wrap in a double layer of plastic wrap and foil. Do not add the icing. Freeze for up to three months. Thaw overnight on the counter, then frost before serving.

If you're using a plastic baby figurine in your cake, wait until after the cake is baked to add it. Once out of the oven, lift up the cake and gently push the baby into the underside of the cake so that it's hidden.

(Emma Christensen is a contributor to TheKitchn.com, a nationally known blog for people who love food and home cooking. Submit any comments or questions to editorial@thekitchn.com.)