In a microwave-safe bowl, warm the milk and butter for 60–90 seconds. Mix in the salt, then leave to cool until just warm, about 120°–130°F (49°–54°C).

In another bowl, stir together the warm water, about 1 tbsp of the granulated sugar, and the yeast. Let stand 5–10 minutes until frothy.

After the yeast is frothy, incorporate the milk-butter mixture, the remaining granulated sugar, eggs, vanilla, and 1 cup of the flour. Combine thoroughly, then keep adding flour while kneading with the dough hook, 1 cup at a time. Add the additional 1/2 cup flour only if the dough is really sticky. Knead 5 minutes on medium speed, or 8–10 minutes by hand, until a very soft dough forms and starts to pull away from the bowl.

Transfer the dough to a floured surface and knead a few times by hand until a smooth, round ball forms. Clean and lightly oil the bowl, put the dough back, and turn to coat. Cover with plastic wrap and let rise in a warm place 1–2 hours until doubled in size.

For the cinnamon filling, mix the brown sugar, granulated sugar, flour, cinnamon, and melted butter until combined. Set aside. Use only half of these amounts if filling just one king cake.

For the cream cheese filling, combine the softened cream cheese, powdered sugar, softened butter, and vanilla in a bowl. Beat well until smooth. Set aside. Use only half of these amounts if filling just one king cake.

Line two baking sheets with parchment paper and set aside.

Once the dough has doubled, punch it down and divide in half (this recipe makes TWO king cakes). Roll one half into a large rectangle on a lightly floured surface, roughly 10x16 inches. Use a pizza cutter to divide the dough in half lengthwise, creating two long rectangles.

Crumble the cinnamon filling or spread the cream cheese filling evenly over the dough, then roll up into two long cylinders, as for cinnamon rolls. Twist the two logs together into a rope braid, transfer to a prepared baking sheet, shape into an oval, and pinch the ends together. Repeat with the remaining dough and filling. Cover lightly with plastic wrap and let rise 30–45 minutes until puffy.

Preheat oven to 350°F (175°C) while the king cake rises. Bake 25–30 minutes until golden brown and baked through; the cake is done when it reaches 190°–195°F (88°–91°C) on a digital thermometer inserted into the center of each loaf. Remove from oven and cool completely before frosting.

Beat the powdered sugar, milk, butter, lemon juice, and vanilla in a medium bowl until smooth. The frosting should be pourable but thick.

Stick a plastic baby figurine into each cake, pushing it down to hide it. Drizzle the frosting over both king cakes, spreading with a spatula if needed for good coverage. Sprinkle with purple, green, and gold sugar while the frosting is still wet, before it sets.

Slice and serve. Whoever gets the slice with the baby gets to be king or queen for the day!