Warm the milk in a small saucepan to no more than 100 degrees F (40 degrees C). Sprinkle the yeast overtop and let stand until the yeast softens and begins to form a creamy foam, about 5 minutes; stir in the sugar.
Whisk the whole wheat flour, all-purpose flour, oatmeal, and salt together in a mixing bowl. Make a well in the center and pour the milk mixture into the well. Stir until the dough has pulled together. Turn the dough out onto a lightly-floured surface and knead until smooth and elastic, about 8 minutes.
Lightly oil a large bowl. Place the dough in the bowl and turn to coat with oil. Cover with a light cloth and let rise in a warm place (80 to 95 degrees F (27 to 35 degrees C)) until doubled in volume, about 45 minutes.
Grease 2 8x4-inch loaf pans. Deflate the dough and turn it out onto a lightly-floured surface. Use a knife to divide the dough into two equal pieces - don't tear it. Shape into dough rounds and let rest for 10 minutes. Form the dough into loaves and place into the prepared pans. Cover with a damp cloth and let rise until doubled in volume, about 40 minutes.
Preheat an oven to 350 degrees F (175 degrees C).
Bake in the preheated oven until the top is golden brown and the bottom of the loaf sounds hollow when tapped, 25 to 30 minutes. Cool in the pans for 5 minutes before turning out onto a wire rack to cool completely.