Dissolve the yeast in warm milk with a pinch of sugar. Let it sit until frothy (about 10 minutes).
In your stand mixer's bowl, mix together the flour, sugar, salt, and ground cinnamon. Add the activated yeast mixture, eggs, and melted butter all at once. Using the dough hook attachment, let the mixer knead the dough on a low setting. This should take about 5-7 minutes, until the dough becomes smooth and elastic.
Place in a greased bowl, cover, and let rise in a warm place for 1 hour or until doubled in size.
Mix brown sugar and cinnamon in a bowl. Set aside.
Roll out the dough into a large rectangle. Spread softened butter over it, then sprinkle the brown sugar-cinnamon mixture evenly. Roll the dough tightly lengthwise. Form it into a circle and pinch the ends together.
Place the shaped dough on a baking sheet or bundt pan, cover, and let rise for 30 minutes.
Preheat your oven to 180°C (350°F). Bake the cake for 30 minutes or until golden brown.
While the cake is cooling, mix powdered sugar, milk, and vanilla extract to make the icing. It should be thick but pourable. Take your granulated sugar and divide it equally into three separate small bowls. Add a few drops of green food coloring to the first bowl and mix thoroughly until all the sugar is uniformly green. Repeat this process with purple food coloring in the second bowl and yellow food coloring in the third bowl. Spread the colored sugars out on a baking sheet or parchment paper to dry for a bit. This prevents the colors from running when you sprinkle them on the icing.
Once your King Cake is iced, sprinkle the colored sugars over it. You can create alternating bands of green, purple, and yellow, or mix them as you like.