1. Whisk milk, 6 tablespoons melted butter, granulated sugar, water, and yeast together in 2-cup liquid measuring cup until yeast dissolves. Whisk flour, salt, and 1 teaspoon cinnamon together in bowl of stand mixer. Using dough hook with mixer on low speed, slowly add milk mixture and mix until dough comes together, about 2 minutes. Increase speed to medium and continue to mix until dough is smooth and elastic, about 10 minutes. (If after 5 minutes dough is still very sticky, add 1 to 2 tablespoons extra flour; dough should clear sides of bowl but stick to bottom.)
2. Transfer dough to lightly floured counter and knead by hand to form smooth, round ball, about 1 minute. Place dough in large, lightly greased bowl, cover tightly with greased plastic wrap, and let rise until doubled in size, about 1 hour.
3. Combine brown sugar and remaining 1 teaspoon cinnamon in bowl. Transfer dough to lightly floured counter and roll into 16 by 12-inch rectangle with long side facing you, Brush remaining 2 tablespoons melted butter over dough, leaving ½-inch border at edges. Sprinkle cinnamon-sugar mixture over butter, leaving ¾-inch border at top edge, and press lightly to adhere. Roll dough away from you into firm cylinder, keeping roll taut by tucking it under itself as you go. Pinch seam and ends closed. If necessary, gently reshape log to be 16 inches in length with even diameter.
4. Grease 10-inch cast-iron skillet. With short side of dough log facing you, cut log in half lengthwise, using bench scraper. Turn dough halves cut sides up and arrange side by side. Pinch top ends together. Lift and place 1 dough half on opposite side of second half. Repeat, keeping cut sides up, until dough halves are tightly braided. Pinch remaining ends together. Twist braided dough into spiral and tuck end underneath. Transfer loaf to prepared skillet, cover with greased plastic, and let rise until doubled in size, 45 to 60 minutes.
5. Adjust oven rack to lower-middle position and heat oven to 325 degrees. Brush loaf with beaten egg; transfer skillet to oven, and bake until loaf is deep golden brown and filling is melted, 45 to 55 minutes, rotating skillet halfway through baking.
6. Using potholders, transfer skillet to wire rack and let loaf cool for 10 minutes. Being careful of hot skillet handle, remove loaf from skillet, return to rack, and let cool to room temperature, about 2 hours, before serving.