Add the milk and butter to a saucepan and melt over medium heat. Do not let the milk boil, the milk should be warm, no more than 105°F. Sprinkle the yeast over it and whisk it a bit. Let it rest for 10 min.
In the mixer bowl, add the eggs, ½ cup of sugar and vanilla extract and mix well until light and fluffy using the whisk attachment.
In another large bowl mix the flour and salt together. Next, add the yeast mixture to the mixer bowl with the egg and mix until it's well blended.
Add half the flour mixture to the mixer, switch to the dough hook attachment, and mix until well combined. Add the remaining flour and continue mixing for another 5 minutes.
Place the dough in an oiled bowl and cover it with plastic wrap. Let it rise until it doubles in size in a warm place.
First, grease a loaf pan 8.5 x 4.5 x 3-inch with butter or spray it with cooking spray. Roll out the dough on a floured work surface until it's a bit wider than the loaf pan and about ¼ inch thick.
In a small bowl mix the ⅓ cup of sugar and cinnamon together and spread the cinnamon sugar evenly over the dough. Roll the dough and pinch the ends sealed.
Place it in the prepared loaf pan with the sealed side down. Let the bread rise again until doubled in size, around an hour.
Preheat oven to 350°F. Brush the bread with the beaten egg and sprinkle some more sugar and cinnamon on top, if preferred.
Bake for about 35 to 45 minutes or until golden brown. The loaf is done baking when an instant-read thermometer inserted into the center of the bread reads 195 F°.