Stir yeast into warm (100°-110°) milk & let stand for 10 minutes.
Whisk flour and water together in microwaveable dish or measuring cup until smooth.
Microwave, whisking every 25 seconds until stiff and pudding-like, 50-75 seconds. Should mound instead of spread when dropped from a spoon. Let cool to ~115° in metal mixing bowl.
In stand mixer bowl, whisk flour paste and milk/yeast together until smooth.
Whisk in egg and yolk until uniform then add flour.
Fit stand mixer with paddle attachment and mix on low until uniformly moist, ~1-2 minutes. Stop and let stand 15 minutes.
Replace paddle with dough hook. Add sugar and salt and mix on medium-low for 5 minutes.
Stop and add softened butter cut up into slices. Mix on medium-low for 2.5 minutes. Scrape down hook and bowl. Continue mixing until until the dough forms into a tight elastic ball.
Place dough in a lightly greased bowl & lightly coat dough with vegetable spray. Cover with plastic wrap.
Let dough rise until doubled in volume, 40-60 mins.Can also be placed in the refrigerator overnight for early morning bakes.
Melt butter over medium heat in a small saucepan. Butter will foam and pop, then start to brown as the solids toast. You can go as light or dark with your browning. It will smell nutty and wonderful. Make right before brushing on dough or make while dough rises & remelt to brush.
On a floured counter, flatten dough & shape into a 12" square. Let rest 10 minutes.
Roll to 1/4" thin forming a rectangle. Let rest 10 minutes.
Roll to 1/8" thin. Let rest 10 minutes.
Roll to 1/16" or thinner. It will be at least 4.5 feet long.
Brush melted brown butter over the entire dough.
Using a pizza cutter, slice dough in half lengthwise. Place one half on top of the other. Repeat. You should now have 4 long layers of dough.
Cut in half making two 2ft pieces. Roll out both pieces to around 4ft again. If dough wants to spring back, let it rest for 10 minutes and try again.
Brush tops with melted brown butter.
Sprinkle brown sugar & cinnamon mixture in a very thin layer over the doughs & pat into place.
Roll each dough tightly into a cylinder. Place seam side down on a parchment lined sheet pan and refrigerate for 15 minutes.
Mark every inch with a very sharp knife to use as your guide. To slice, hold a strand of dental floss taut and slide underneath the cylinder, stopping at the first mark. Cross ends of floss over each other and pull.
Transfer each slice, cut side down, to your parchment lined baking sheet, cover with plastic wrap and let chill in fridge for 45-55 minutes.
Preheat oven to 375°F
Place slices on new parchment paper in a circle for traditional king cake shape or space out for individual cakes.
Bake 25-30 minutes, turning halfway through for even color until golden brown. Top with Brown Butter Cream Cheese Icing while warm.