Prep the yeast mixture: 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.
Combine sugar and eggs: 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.
Start the dough: 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 Flour: 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.
First Rise: 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.
Roll the dough: 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.
Make the loaf: 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.
Second Rise: Place it in the prepared loaf pan with the sealed side down. Let the bread rise again until doubled in size, around an hour.
Prep for Baking: Preheat oven to 350°F. Brush the bread with the beaten egg and sprinkle some more sugar and cinnamon on top, if preferred.
Bake: 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°.