If this if your first time working with yeast or baking bread, I recommend reading through the steps below at least once before you begin.

1. Gather and prep. Measure out all of your ingredients and have them at arm's reach. Grate the zest of the orange and lemon. Allow the butter to soften, and the milk and eggs to come to room temperature.
2. Combine the dry ingredients. Whisk together in a bowl the all-purpose flour, instant dry yeast, and nutmeg.
3. Cream the butter, sugar, and salt. In the bowl of a stand mixer with the paddle attachment (or in a large bowl with a hand mixer) beat the butter, sugar, and salt at medium-high speed until lightened in color and smooth, about three minutes. Scrape the bowl and the beater(s).
4. Add the eggs, flavorings, and milk. If the milk feels cold to the touch, warm it briefly over low heat on the stove or for 20 seconds in the microwave. Beat the citrus zest and vanilla extract into the butter mixture. Add the eggs one at a time beating at medium speed, letting the first blend in completely before adding the second. Scrape the bowl and beater to keep the mixture as smooth as possible. Add the milk and beat for a minute more (the mixture may appear lumpy). 
5. Add the dry ingredients. On low speed, add the flour mixture a little at a time, scraping down the bowl and beater often to create a smooth dough. If using a handheld mixer and kneading by hand, mix in the last third of the flour with a wooden spoon.
6. Knead the dough. Scrape all the dough off the mixer paddle and switch to the dough hook attachment. Beat at medium speed or knead the dough by hand until it is smooth, supple, and elastic when gently stretched. This takes about 10 minutes in a stand mixer. By hand, it will take several minutes longer. The mixer bowl or kneading surface should be perfectly clean, and the dough should not be sticky (add a little more flour as you begin to knead if it is).
7. Proof. Shape the dough into a large round, brush the top with oil, and let it rise in a covered, oiled bowl until it doubles in size. This takes at least two hours. You can also proof the dough in the refrigerator overnight (about 12 hours). Let it come to room temperature before moving on to the next step, which can take several hours.
8. Mix the filling. When the shaped dough has almost doubled in size, mix together the filling ingredients. You can mix them ahead of time and store the filling covered.
9. Roll the dough and shape the cake. Press the air out of the risen dough. On a lightly floured surface, roll it into a large rectangle, about 12" x 24" in size. If the dough begins shrinking back as you roll, cover it with plastic wrap, let it rest for 20 minutes, and then continue rolling.
10. Add the filling. Brush the top of the dough with oil or melted butter. Cut it lengthwise into three like-size strips. Then spread the filling evenly among the strips leaving space at the edges. Starting with a long side, roll each strip up and pinch the edges closed to seal in the filling.
11. Braid. Lay the strips of dough side-by-side and braid them together tightly. Once braided, press and roll the braid gently to lengthen it a little. Start with your hands in the center of the braid and work your way to the ends. Transfer the braid to a parchment-ling baking pan and shape it into a circle or oval. Pinch the open ends together well to seal them off and tuck the closure underneath, if possible.
12. Final proof. Cover the shaped dough with a tea towel or plastic wrap and let it proof (rise) until it doubles in size. This can take an hour or up to two hours. Judge with your eyes and not the clock.
13. Preheat and bake. When the dough has nearly doubled in size, preheat your oven to 350° F. Brush the cake with melted butter or oil, and bake for 18 to 22 minutes. The king cake is done when the top is golden brown and it sounds hollow when tapped. The internal temperature of the cake should be 190° F to 195° when measured with a probe (meat) thermometer. 
14. Cool and decorate. Cool the cake completely before decorating. Push a baby figure, bean, or pecan half into the bottom of the cake when no one is looking. Whisk the icing ingredients until smooth and drizzle evenly over the top of the cake. Sprinkle on the purple, green, and gold (yellow) colored sugar or sprinkles. 
15. Store covered. King cakes stay moist for a couple of days when stored airtight at room temperature. King cakes with perishable fillings should be refrigerated. If you do chill the cake, allow it to come to room temperature before serving. Cold constricts flavors and toughens the texture of bread and cakes.