Whisk the yeast with the warm milk in the bowl of a stand mixer until dissolved. Add the 6 tablespoons of bread flour and the honey and, using the paddle attachment, mix on low speed until fairly smooth (there will still be a few lumps), 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 3/4 cup of the remaining bread flour, the 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 the speed to medium-high and begin adding 4 tablespoons of the butter 1 tablespoon at a time, mixing well between additions. Continue to knead until the dough forms a slack ball (it will ride the dough hook, be tacky, and not slap the bottom of the bowl, but it should generally come together into a loose mass), 2 to 3 minutes. If the dough doesn’t come together, continue kneading while adding up to 1/4 cup of the reserved bread flour, until it does.
Grease a large bowl with 1/2 tablespoon of the remaining butter and transfer the dough to the bowl, turning it over in the bowl to coat with butter. Cover the bowl with a piece of plastic wrap or damp kitchen towel and place the bowl in a draft-free spot until the 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 1/4 cup of bread flour (if you used the bread flour in the dough, dust your work surface with more 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 1/4-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 11/2-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.
Heat the oven to 375°F. 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.
To make the icing: 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 of the sugar into each of 3 resealable 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 (if you don’t have purple, make it yourself: measure 2 drops of red and 2 drops of blue food coloring onto a spoon and mix with a cake tester or toothpick until combined). 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. I like to alternate colors every 21/2 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 (unscented trash bags work well) for up to 2 days.