In a large bowl, dissolve the yeast in warm water. Add the 1/2 cup sugar, butter, milk, yolks, salt, lemon peel, nutmeg, and 2 cups of flour. Beat until smooth. Stir in enough of the remaining flour to form a soft dough. The dough will be sticky.
Turn the dough out onto a floured surface. Knead until smooth or elastic, about 6-8 minutes. Place in a greased bowl, turning once to grease the top. Cover and let rise in a warm place until doubled in size, about an hour.
Punch down dough. Turn onto a lightly floured surface and roll into a 16x10-inch rectangle. In a small bowl, combine the cinnamon with the 1/4 cup sugar. Sprinkle over the dough to within 1/2-inch of the edge. Roll up jelly-roll style, starting on the long side. Pinch the seam to seal. Place seam-side down on a greased baking sheet - pinch ends together to form a ring. Cover and let rise until doubled, about an hour.
Preheat oven to 375°F. Brush the cake with the beaten egg. Bake for 25-30 minutes or until golden brown. Cool completely on a wire rack.
For the glaze, combine the powdered sugar, lemon juice, and enough water to achieve the desired consistency. Spread or pour over the cooled cake. Sprinkle with the colored sugars.
Notes
King Cake typically lasts up to five days when stored in an airtight container at room temperature. It can also be frozen for up to three months.
A small plastic baby figurine is often hidden inside the cake. Tradition dictates that whoever finds the baby in their slice of cake is responsible for buying the next King Cake or hosting the next Mardi Gras celebration.
Want to add a plastic baby to your cake? After baking and once the cake has cooled, and before decorating, make a small slit in the bottom of the cake and place the baby into the hole.