In a large bowl, whisk the flour, sugar, salt, and yeast, then set it aside.
Warm the milk and butter in the microwave until the butter melts and the mixture feels warm but not hot, around 100 to 110°F.
Pour the warm milk mixture into the dry ingredients, add the egg, and mix until a soft dough forms using a mixer or a sturdy spatula.
Knead the dough for 5 minutes by hand or in a mixer until it’s smooth and a bit tacky but not too sticky.
Shape the dough into a ball, cover it with a towel or plastic wrap, and let it rest for 30 minutes at room temperature.
Roll the dough into a 15×12-inch rectangle and spread the softened butter all over the top.
Stir together the brown sugar and cinnamon, then sprinkle it evenly over the buttered dough.
Slice the dough into six long 2-inch-wide strips, roll one into a spiral, place it in the center of a greased 9-inch pan, and wrap the remaining strips around it one at a time to form one big swirl.
Cover loosely and let the dough rise for 1 to 2 hours until doubled in size.
Preheat the oven to 350°F, then bake the cake for 30 to 35 minutes until golden brown on top.
Cool the cake in the pan for 10 minutes while you whisk together the icing, then drizzle it all over the warm cake and serve.