Make the dough. In the microwave, heat melted butter and milk in 30 second bursts until lukewarm to the touch (110F), but not hot or steaming. Sprinkle yeast over the liquid and let sit for 5 minutes until foamy. Add egg directly on top and beat lightly with fork. Set aside.
In a large bowl, add dry ingredients. Make a well in the center and add the wet ingredients. Mix with a spatula until a ball forms. Knead lightly until smooth. The dough will be sticky.
Let the dough rise. Cover and transfer the dough to a dark, warm place. Leave undisturbed until doubled in size, about 1 hour. Deflate the dough by punching it down (push down the middle of the dough with your fist until it deflates). Knead a few times. Cover until doubled in size again, about 30-45 minutes. (Yes, the dough needs to rise twice).
Make donuts. Lightly flour the surface of a cutting board. Using a bench scraper or knife, divide dough into 16 equal pieces. Take one piece and roll out (with your hands) into a long strip, about 10 inches long. Twist one end upwards while simultaneously twisting the other end downward. Hold the ends and lift the dough off the surface, in a U-shape. The donut should naturally twist. If you need to, give it a little extra twist to help it along. Pinch the ends together and tuck under each other. Transfer to a sheet pan lined with parchment paper. Repeat until you have 16 donuts.
Let the twisted dough rise. When all the donuts are twisted, cover. Leave undisturbed in a warm, draft-free place until doubled in size, about 15-30 minutes.
Deep fry. Heat a dutch oven with 2-3 inches of cooking oil to 350F. Fry donuts until puffy and golden, about 2-3 minutes each. Make sure to flip the donuts. Do not crowd the pan!
Dust with cinnamon sugar. In a large shallow bowl, combine the cinnamon and brown sugar. Mix well. Transfer hot Korean donuts to the cinnamon sugar mixture. Roll the donuts and spoon extra cinnamon sugar on top. Flip around and around until they are thoroughly coated. Enjoy!
Eat immediately! Korean donuts taste best when they are still warm.