In the bowl of a stand mixer, combine warm milk, 1 tsp of granulated sugar, and active dry yeast. Stir gently and let sit for 5-10 minutes until foamy.

Add the remaining granulated sugar, 3 1/2 cups flour, salt, and eggs to the yeast mixture. Attach the dough hook and mix on low speed until a shaggy dough forms, about 2-3 minutes.

With the mixer still on low, add the softened butter, one tablespoon at a time, allowing each to incorporate before adding the next. Once all butter is added, increase speed to medium-low and knead for 8-10 minutes until the dough is smooth, elastic, and pulls away from the sides of the bowl.

Lightly grease a large bowl. Transfer the dough to the bowl, turning once to coat. Cover with plastic wrap and let rise in a warm place for 1 to 1.5 hours, or until doubled in size.

While the dough is rising, prepare the cream cheese filling. In a medium bowl, combine softened cream cheese, 1/4 cup granulated sugar, vanilla extract, and 1 tbsp all-purpose flour. Mix with a spatula or hand mixer until smooth and well combined. Set aside.

Once the dough has doubled, gently punch it down. Lightly flour a clean work surface and turn out the dough.

Using a rolling pin, roll the dough into a large rectangle, approximately 12x24 inches.

Spread the cream cheese filling evenly over the entire surface of the dough, leaving a 1-inch border along one of the long edges.

Starting from the long edge opposite the border, tightly roll the dough into a log. Pinch the seam closed to seal.

Carefully transfer the log to a parchment-lined baking sheet. Bring the two ends together to form an oval or ring shape, pinching them firmly to seal.

If using a plastic baby, insert it into the bottom of the cake, ensuring it's fully hidden.

Cover the king cake loosely with plastic wrap or a clean kitchen towel and let it rise in a warm place for another 30-45 minutes, or until puffy.

Preheat oven to 375°F (190°C). In a small bowl, whisk together the egg and water for the egg wash. Gently brush the entire surface of the risen king cake with the egg wash.

Bake for 25-30 minutes, or until golden brown and cooked through. If the cake is browning too quickly, you can loosely tent it with foil.

Remove from the oven and transfer to a wire rack to cool completely before glazing.

For the Glaze: In a medium bowl, whisk together 2 cups powdered sugar, 3-4 tbsp milk, and 1/2 tsp vanilla extract until smooth and pourable. If too thick, add more milk 1 tsp at a time; if too thin, add more powdered sugar.

Once the cake is completely cool, pour and spread the glaze evenly over the top of the king cake, allowing some to drip down the sides.

For the Colored Icing: Divide 1 cup powdered sugar into three small bowls. Add 1-2 teaspoons of milk to each bowl (or enough to make a thick but flowable icing) and mix. Add a few drops of purple food coloring to one, green to another, and yellow to the third. Mix well.

Drizzle the colored icings over the plain glaze in alternating stripes or patterns.

Immediately sprinkle with purple, green, and gold sprinkles (or sanding sugar) before the icing sets.

Let the icing set for at least 30 minutes before serving.