Combine the warm milk and oats in the bowl of a stand mixer. Let sit for 30 minutes to soften the oats.
To the mixer bowl with the oats and milk, add the butter, honey, warm water, yeast, salt, all-purpose flour and whole wheat flour. Mix using the paddle attachment until everything is combined.
Switch to the dough hook. Knead the dough on low speed for about 8 minutes, until dough pulls away from the sides of the bowl. The dough should feel slightly sticky when you touch it, but a lot of dough shouldn't stick to your finger. If the dough is very sticky, add a little more flour, 1 to 2 tablespoons at a time.
Cover the bowl with a clean kitchen towel and let the dough rise for 60-90 minutes, until it has doubled in size.
Grease a 9 x 5-inch or 8 ½ x 4 ½-inch metal or glass bread pan (see note). Transfer the dough to a lightly floured work surface. Knead it a few times with your hands to release any large air bubbles. Shape dough into a ball. Then use your hands to gently form the ball into a loaf shape. (To do this, cup the dough between your hands as you rotate it on the counter, shaping it into an oval loaf and tucking the ends under as needed.) Put the dough in your greased bread pan, seam side down.
Cover with a clean kitchen towel and let the dough rise in the bread pan until the top of the loaf is just above the top of the pan. This should take about 20 to 30 minutes. Meanwhile, preheat oven to 350° F.
Right before baking, add the topping: In a small bowl beat together the egg white and water with a fork. Brush over the top of the loaf (you will not use all of the egg wash), then sprinkle on the 2 tablespoons oats.
Bake the bread for 45-55 minutes (see note), until the center of the loaf registers 200° F on an instant-read thermometer. Check on the bread partway through baking and if the top is browning too much, cover the top of the loaf loosely with foil.
Let the bread cool for 15 minutes in the pan and then remove it to a wire rack to finish cooling. The bread will slice easiest when it is cool. Store in an airtight container at room temperature for up to 3 days or in the freezer for up to 3 months.