Soften the Oats: Place the rolled oats and softened butter in the bowl of a stand mixer or a large mixing bowl. Bring the milk to a simmer, then pour it over the oats and butter. Stir to combine and leave the mixture for about 1 1/2 hours, or until it has cooled to room temperature. Combine the Dough Ingredients: Add the honey, warm water, instant yeast, and kosher salt to the cooled oat mixture. Stir thoroughly to combine.
Add Whole Wheat Flour: Add the whole wheat flour and mix on medium-low speed until incorporated.
Add All-Purpose Flour: Gradually add the all-purpose flour, a little at a time, until the dough begins pulling away from the sides of the bowl. Use only enough flour to create a soft dough that is not overly sticky; avoid adding too much flour.
Knead the Dough: Knead the dough for a few minutes, scraping down the sides and bottom of the bowl as needed, until the dough becomes smooth and elastic.
First Rise: Transfer the dough to a greased bowl and cover it with a dish towel or plastic wrap. Let it rise for about 1 hour, or until it has doubled in size.
Divide the Dough: Punch down the risen dough, then divide it into two equal portions.
Shape the Loaves: Shape each portion into a loaf and place the two loaves in lightly greased loaf pans.
Apply the Topping: Combine the egg white and 1 tablespoon water to make the egg wash. Gently brush the tops of both loaves with the egg wash, then sprinkle a small handful of dry oats over each loaf.
Second Rise: Lightly grease a piece of plastic wrap and gently place it over the loaves. Let them rise in a warm place for 45 minutes to 1 hour, or until the dough has risen about 1 inch above the loaf pans. Carefully peel away the plastic wrap.
Bake the Bread: Bake the loaves at 350°F for 30–40 minutes. They are ready when lightly tapping the tops produces a hollow sound, or when an instant-read thermometer registers 200°F.
Cool the Loaves: Let the baked loaves cool on a wire rack for 15 minutes. Turn them out of the pans and allow them to cool completely before serving.