Place flour, white sugar, salt, and yeast into the bowl of a stand mixer. In a bowl, whisk together egg, melted butter, warm milk, and vanilla extract; pour into the flour mixture. Using dough hook attachment, mix on low until the flour is moistened and a dough forms, then increase speed to medium, and continue kneading until smooth and elastic, 4 to 5 minutes.
Place dough into a greased bowl, cover, and allow to rise in a warm place until doubled in bulk, about 1 1/2 hours.
Punch down dough, turn out onto a floured work surface, and divide into 2 equal pieces. Roll each piece into a 1/3 inch thick rectangle. Brush each square with melted butter. Mix together cinnamon and brown sugar in a small bowl, then sprinkle evenly over the two rectangles.
Roll each firmly into a log, pinch the ends closed, and tuck them underneath. Place each into a greased, glass loaf pan. Cover, and allow to rise in a warm place until doubled in bulk, about 45 minutes.
Preheat oven to 350 degrees F (175 degrees C).
Brush the tops of the loaves with melted butter, then bake in preheated oven until loaves are golden brown, and sound hollow when tapped, about 30 minutes.