In the bowl of a stand mixer, combine the warm milk and oats. Let sit for 30 minutes to allow the oats to soften.
Add the butter, honey, warm water, yeast, salt, all-purpose flour, and whole wheat flour to the bowl with the oats and milk. Using the paddle attachment, mix until well combined.
Switch to the dough hook and knead on low speed for about 8 minutes, until the dough pulls away from the sides of the bowl. It should feel slightly sticky but not overly wet. If the dough is too sticky, add additional 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, or until doubled in size.
Grease a 9x5-inch or 8½x4½-inch metal or glass bread pan. Lightly flour a work surface and transfer the dough onto it. Knead a few times to release large air bubbles, then shape the dough into a ball. Gently form it into an oval loaf by cupping it between your hands and tucking the ends under as needed. Place the dough into the greased bread pan, seam side down.
Cover with a clean kitchen towel and let the dough rise in the pan until the top of the loaf is just above the rim, about 20-30 minutes. Meanwhile, preheat the oven to 350°F.
Prepare the topping: In a small bowl, whisk together the egg white and water with a fork. Lightly brush the mixture over the top of the loaf (you won’t need all of it), then sprinkle 2 tablespoons of oats on top.
Bake for 45-55 minutes, or until the center of the loaf registers 200°F on an instant-read thermometer. If the top begins to brown too quickly, loosely cover it with foil partway through baking.
Let the bread cool in the pan for 15 minutes before transferring it to a wire rack to cool completely. For the best slices, wait until the bread is fully cooled before cutting.
Storage: Keep in an airtight container at room temperature for up to 3 days, or freeze for up to 3 months.