Combine 1/2 cup all-purpose flour and yeast in a bowl; stir in warm water. Let rest until doubled in volume, 30 to 40 minutes.
Heat 1/2 cup milk in a small saucepan over medium heat until it begins to bubble. Combine milk, whole wheat flour, 1/4 cup sugar, 2 tablespoons butter, and 1/2 teaspoon salt in a bowl. Allow to cool slightly. Add yeast mixture; stir to combine.
Stir in additional all-purpose flour until dough forms a ball and pulls away from the bowl. Transfer to a floured work surface; knead until smooth and elastic. Place dough in a greased bowl; cover with plastic wrap and let rise until doubled in volume, 45 minutes to 1 hour.
Punch dough down; let rest 10 minutes more.
Roll dough out on a floured work surface into a 12x20-inch rectangle. Spread 2 tablespoons melted butter on the dough. Combine 1 cup confectioners' sugar, cocoa powder, and cinnamon in a bowl; sprinkle over the dough.
Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper.
Roll dough up jelly-roll fashion. Shape roll into a circle on the prepared baking sheet. Slice roll into 16 slices, cutting only 3/4 of the way through. Twist every other slice flat to the inside of the circle; twist the other slices flat to the outside of the circle. Cover with plastic wrap; let rise until doubled in volume, about 30 minutes.
Bake in the preheated oven until golden, about 20 minutes. Remove from baking sheet; let cool.
Combine 1 cup confectioners' sugar, 2 tablespoons milk, and 2 tablespoons melted butter. Drizzle glaze over coffee cake.