Place 1 cup of rolled oats and the softened butter in a mixing bowl. Pour the warm milk over the top and stir to combine. Let the mixture cool to room temperature, about 1–2 hours.
In a separate large mixing bowl, combine the warm water and yeast. Stir and let sit for 3–5 minutes, until the yeast begins to foam.
Add the honey and salt to the yeast mixture and mix until combined.
Add the cooled oatmeal mixture and 1 egg. Mix with a hand mixer or stand mixer until combined.
With the mixer on low speed, gradually add the flour. Mix until a dough forms.
Knead the dough for a few minutes, until smooth and elastic.
Place the dough in a greased bowl. Cover and let rise for about 1 hour, or until doubled in size.
Gently punch down the dough and transfer it to a lightly floured surface. Knead briefly, then shape into a loaf and place in the prepared loaf pan.
For the topping, whisk together the remaining egg and 1 tablespoon water. Brush the egg wash over the loaf and sprinkle with the remaining ¼ cup rolled oats.
Cover the loaf loosely with greased plastic wrap and let rise for 45–60 minutes, or until doubled in size. Do not let the dough rise over the top of the pan.
While the dough is rising, preheat the oven to 375°F.
Remove the plastic wrap and bake for 30–40 minutes, until the top is golden brown and the loaf sounds hollow when lightly tapped.
Cool in the pan for 10 minutes. Carefully remove the bread from the pan and transfer it to a wire rack to cool completely.
Slice and serve.
Notes
Store the bread in an airtight container at room temperature for up to 3 days.