Sprinkle the active dry yeast over the warm milk and stir well. Let it sit for 3 to 5 minutes until the yeast dissolves and activates.
In the bowl of a stand mixer, combine the flour, sugar, and salt. Mix to blend evenly.
Add the activated yeast mixture and eggs to the flour mixture. Using the dough hook attachment, mix on medium-low speed until the dough begins to come together. Increase to medium-high and knead until smooth and elastic.
Once the dough passes the windowpane test, add the softened butter and continue kneading on medium speed until fully incorporated.
Shape the dough into a ball, place in a clean bowl, cover and let rise in a warm place until doubled in size, about 1 hour.
Mix the brown sugar and cinnamon in a small bowl. Set aside.
Optional: Whisk softened butter until pale and fluffy. Set aside.
Line a sheet pan with parchment paper.
After dough has doubled, gently punch down, fold edges toward center and transfer to a floured surface.
Roll dough into 13×22 inch rectangle, dust lightly with flour if needed.
Spread softened butter on dough surface leaving ½ inch borders.
Sprinkle cinnamon sugar mixture evenly and press firmly into butter.
Cut dough lengthwise in half creating two rectangles.
Roll top rectangle upward from middle cut edge to clean edge, seal seam and ends.
Repeat rolling with bottom rectangle downward and seal.
Twist two dough logs together, connect ends to form a circle.
Transfer braided dough to prepared pan, shape round or oval.
Cover and proof in warm area until nearly doubled, about 45 minutes to 1 hour.
Preheat oven to 350°F (180°C).
Bake king cake for 30 minutes until crust is golden and internal temp is 190°F.
Cool cake completely before icing and decorating.
Mix cream cheese in stand mixer with paddle attachment on medium speed.
Add powdered sugar and mix on medium-low until combined.
Add vanilla extract and mix on medium speed 3 to 5 minutes until smooth and creamy.
Optionally insert king cake baby in bottom of cooled cake.
Spread cream cheese icing evenly over the top.
Sprinkle yellow, green, and purple sugars alternating colors over icing.
Slice and serve. Warm slices briefly in microwave before serving for best texture.