In a small bowl, stir the oats into the heated milk. Mix in the butter, brown sugar, and salt.
In a large bowl, sprinkle yeast over the warm water. Let sit for a minute before stirring to mix. Let rest until frothy, 5-10 minutes. Once the oat and milk mixture is below 115 degrees F, add the the frothy yeast. Stir in 1 cup flour until well combined. Slowly incorporate remaining flour until a smooth dough forms. Cover and let rest 15 minutes.
Lightly flour work surface. Knead dough until smooth and elastic, about 10 minutes. Grease large bowl and add dough, turning to coat. Cover and let rise until doubled, about 1 hour.
Grease two 9 inch round pans. Divide the dough into 2 equal pieces. On lightly floured surface, shape each piece into a round loaf, tucking and sealing the edges under to create a smooth top surface. Place each loaf in prepared pan. Cover and let rise until doubled, about 30 minutes.
While rising, preheat oven to 375 degrees F. In a small bowl, beat egg with the milk. Brush the milk mixture over the risen loaves. Sprinkle with oats. Bake in preheated oven until golden on top and hollow sounding when tapped, about 25 minutes. Let rest in pan about 5 minutes before removing to wire rack to cool.