Scald the milk. Stir in 2 cups rolled oats, brown sugar, salt and butter. Remove from the heat and cool to lukewarm.
Sprinkle yeast on the warm water; stir to dissolve.
Add the milk mixture and 2 cups flour to the yeast. Beat with a mixer on medium speed, scraping the bowl occasionally, for about 2 minutes. (You can also beat with a spoon until the batter is smooth.)
Add enough remaining flour, a little at a time, first with a spoon and then with your hands, to make a soft dough that leaves the sides of the bowl. Turn onto a floured surface and knead until dough is smooth and elastic, about 8-10 minutes. Place in a lightly greased bowl; turn dough over to grease the top. Cover and let rise in a warm place until doubled, about 1 to 1½ hours. Punch down and let rise again until nearly doubled, about 30 minutes.
Turn onto a floured surface and divide in half. Round up to make 2 balls. Cover and let rest 10 minutes. Shape into loaves and place in greased 9x5x3" loaf pans. Let rise until almost doubled, about 1 hour 15 minutes. Brush the tops of the loaves with egg white beaten with water and sprinkle with rolled oats.
Bake in a 375° oven for about 40 minutes. (If the bread starts to brown too much, cover loosely with a sheet of aluminum foil after baking 15 minutes.)