In the bowl of a stand mixer, lightly whisk together the warm water, yeast, and 1 tablespoon of sugar. Allow the mixture to sit for about 10 minutes until it is foamy.
Add the melted butter, milk, egg, egg yolk, vanilla extract, cinnamon, nutmeg, salt, remaining ¼ cup sugar, and one cup of flour. Use a wooden spoon to mix the ingredients until combined.
Add the last 2 cups of flour, one cup at a time, and mix on medium speed (4-5 on a stand mixer) with the paddle attachment after each addition.
Once a shaggy dough has formed, switch to the bread hook and knead the dough on medium speed for 8 to 10 minutes until the dough is pulling away from the side and is smooth and elastic. If the dough is too sticky add up to ½ cup more flour, 2 tablespoons at a time, until desired consistency is reached.
Shape the dough into a ball and place it into a clean lightly oiled bowl. Cover the bowl with plastic wrap and allow the dough to rise in a warm spot for about an hour or until doubled in size.
Once the dough has doubled in size, punch it down, and turn it out onto a lightly floured surface. Roll the dough out into a 16 x 10-inch rectangle with a lightly floured rolling pin.
Slice the dough in half the lengthwise with a pizza cutter. Mix the brown sugar, cinnamon, melted butter, and salt together in a small mixing bowl with a fork. Spread the filling out over the dough with your fingers leaving a small rim along the long edges.
Roll each section of dough into a log like you would a cinnamon roll. Carefully twist the logs together into a two-strand rope. Shape the twisted dough into a circle and pinch the ends of the rope together.
Place the pastry on a parchment paper lined baking sheet, loosely cover it with plastic wrap, and allow it to rise in a warm spot for about an hour or until doubled in size.
Preheat your oven to 350°F. Bake the king cake for 25 to 30 minutes or until golden brown and done in the center. Allow the cake to cool completely.
In a medium mixing bowl, whisk together the powdered sugar, lemon juice, and milk. Tuck the plastic baby into one of the folds of the cake and spoon the glaze over the cake. Finish the cake by sprinkling green, gold, and purple colored sugars in sections on top of the glaze before it sets completely. Slice and enjoy!