In a small bowl, stir together warm milk, sugar, and yeast. Set aside for 5-10 minutes.
In a medium bowl, stir together yogurt, melted butter, egg yolks, vanilla extract, and orange zest. Add the yeast mixture.
In the bowl of a stand mixer using the dough hook attachment, add the flour, nutmeg, and salt. Stir.
Add in the yogurt-yeast mixture. Stir until combined, pushing down sides a few times.
Knead using the dough hook attachment for about 5 minutes.
Meanwhile, oil the inside of a mixing bowl.
Add the dough to the mixing bowl, and cover with a towel. Set aside until it doubles in size, about 90 minutes.
When ready, punch down the dough and roll into a large 18x12 inch rectangle. Spread with the softened butter, brown sugar, and cinnamon. Roll and cut into 12 pieces.
Roll each piece into a circle and place them on a parchment-lined baking sheet. Set aside for 25-45 minutes - until they have just about doubled in size.
Preheat the oven to 375*F.
Bake the mini king cakes for 12-16 minutes until lightly golden. Let cool.
Stir together the powdered sugar, a tablespoon of milk, and vanilla. Add more milk if needed to make the glaze.
Drizzle the glaze over the cooled king cakes and sprinkle with colored sugar. Enjoy!
Notes
*Be sure your yeast is alive and bubbly before doing anything else.*Make sure you roll your dough really wide and pretty thick. You don't want the dough too thin when you are about to spread it with the butter for filling. This will prevent tearing. You want it to be as wide as you can make it so you can cut it into long enough pieces to shape into a circle.*Based on this recipe from Southern Living