Heat water, milk, butter and brown sugar over medium heat until simmering. Remove from heat and add oats. Stir to combine and set aside to cool to room temperature.
Whisk bread flour, whole wheat flour, salt and yeast together in the bowl of a stand mixer. Add cooled oatmeal and milk mixture. Using the dough hook, mix on medium low speed until ingredients are fully combined, scraping down the sides of the bowl as necessary.
Increase speed to medium and knead until bread is smooth and clears the bottom and sides of the bowl, about 5-6 minutes.
Cover bowl and let rise at room temperature until dough has doubled in size, about 1 ½ to 2 hours.
Turn dough out onto a clean work surface. Deflate and shape into a round disk about 2 inches high and place in a greased 9 inch round cake pan. Cover with plastic wrap and let rise at room temperature until the loaf is puffy and slowly springs back when you press it with your finger, about 45 minutes.
Meanwhile, preheat oven to 375 degrees. Whisk egg and milk together. Brush the top of the loaf with egg and milk and sprinkle with oats. Bake 40-50 minutes or until the loaf is golden brown and the internal temperature of the reaches 205-210 degrees. Cool completely before serving.