Add the 1 cup milk and ½ cup unsalted butter to a saucepan and melt over low heat. Take the pan off the heat and let it cool for about 5 minutes, until it reaches 105°F. It will be hotter than that when the butter finishes melting, and anything above 120°F will kill the yeast.
Sprinkle the 2¼ teaspoons active dry yeast over the warm milk, whisk it in, and let it rest for 10 minutes until foamy. If it doesn't foam, the yeast is dead and you'll need to start again with fresh yeast.
In the mixer bowl, whisk the 2 large eggs, ½ cup granulated sugar, and 1 teaspoon vanilla extract with the whisk attachment until light and fluffy.
In a separate bowl, mix the 3½ cups all purpose flour and 1 teaspoon salt together.
Add the yeast mixture to the egg mixture and blend well. Add half the flour mixture, switch to the dough hook, and mix until combined. Add the remaining flour and mix for another 5 minutes, until the dough is soft, slightly tacky, and pulling away from the bowl.
Place the dough in an oiled bowl, cover with plastic wrap, and let it rise in a warm place until doubled in size, about 1 hour.
Grease an 8.5 by 4.5 inch loaf pan. In a small bowl, mix the ⅓ cup granulated sugar, 2 tablespoons ground cinnamon, and 1 tablespoon all purpose flour together.
Roll the dough out on a floured surface until it's slightly wider than the loaf pan and about ¼ inch thick. Spread the cinnamon sugar mixture evenly over the dough, leaving the last inch along the far edge bare.
Roll the dough up tightly, keeping tension as you go, then pinch the seam and both ends sealed. Place it in the loaf pan seam side down.
Cover loosely and let it rise again until doubled, about 1 hour. It should crest just above the rim of the pan.
Preheat the oven to 350°F. Brush the loaf with the beaten egg and sprinkle a little extra cinnamon sugar on top if you like.
Bake for 35 to 45 minutes, until golden brown and an instant read thermometer inserted into the center reads 195°F. If the top browns too quickly, tent it loosely with foil for the last 10 minutes.
Cool in the pan for 10 minutes, then turn out onto a wire rack. Let it cool for at least 1 hour before slicing.