For the cake: Whisk yeast with warm milk in bowl of a stand mixer until dissolved. Add 6 tablespoons bread flour and honey and mix on low speed using paddle attachment until fairly smooth, 30 seconds to 1 minute, scraping the bottom and sides of the bowl as necessary. Cover with plastic wrap and let rise until doubled in volume, about 20 minutes.
Once the dough has doubled, add ¾ cup of remaining bread flour, cake flour, eggs, egg yolk, sugar, cinnamon, vanilla and almond extracts, and salt. Mix on low speed until combined, then switch to a dough hook, increase the speed to medium, and beat until smooth, about 2 minutes. Increase speed to medium-high and begin adding 4 tablespoons of butter 1 tablespoon at a time, mixing well between additions. Continue to knead until the dough forms a slack ball, 2 to 3 minutes. If the dough doesn’t come together, continue kneading while adding up to ¼ cup of the reserved bread flour, until it does.
Grease a large bowl with ½ tablespoon of remaining butter and transfer dough to the bowl, turning it over in the bowl to coat with butter. Cover bowl with plastic wrap or damp kitchen towel and place in a draft-free spot until dough has doubled in size, about 1 hour.
Line a rimmed baking sheet with parchment paper and grease the parchment paper with the remaining butter. Generously flour your work surface using the remaining ¼ cup of bread flour.
Turn the dough out onto the work surface and sprinkle the top with some flour. Use your hands to press and flatten it into a rectangle. Using a rolling pin, roll the dough into a ¼-inch-thick strip that is about 24 inches long by about 6 inches wide. Starting with one of the long sides, roll the dough on top of itself, making a long, thin baguette-shaped length. Pinch the edge to the body of the dough to seal, turn the dough so it lies horizontally on your work surface, and gently roll it on your work surface to even out any bulges and create a somewhat consistent 1½-inch-wide rope. Bring the two ends of the dough together and pinch them into one another to seal. Carefully transfer the dough oval or circle to the prepared sheet pan. Cover with a piece of plastic wrap or a damp kitchen towel and set in a warm, dry spot to rise until doubled, about 1 hour.
For the egg wash: Heat the oven to 375 degrees. To make the egg wash, whisk the egg and the milk together in a small bowl. Brush the egg wash over the top and sides of the dough, and bake the king cake until golden and cooked through, 25 to 30 minutes. Immediately after removing the cake from the oven, make a small slit in the bottom of the cake and insert the baby figurine (if using). Set on a rack to cool completely.
For the icing and decoration: While the cake cools, make the icing. Whisk the confectioners’ sugar, corn syrup, milk, and vanilla together in the bowl of a stand mixer on low speed until smooth and completely incorporated. Cover the bowl with a damp kitchen towel until you are ready to glaze the cake.
To make the colored sugar, measure 1 cup sugar into each of 3 re-sealable quart-size plastic bags. Add 4 drops of green food coloring to one bag, 4 drops of gold or yellow food coloring to another bag, and 4 drops of purple food coloring to the last bag. Seal each bag and then vigorously shake to combine the sugar and food coloring.
Spoon the icing over the cooled cake. Immediately after icing, decorate with the tinted sugar. David likes to alternate colors every 2½ inches, but you can also divide the cake into 3 sections and apply one color to each section. Slice and serve immediately or store in a cake box or on a baking sheet placed within a large plastic bag for up to 2 days.