Warm milk in a small saucepan to no more than 100 degrees F (40 degrees C); sprinkle yeast over top. Let stand until yeast softens and begins to form a creamy foam, about 5 minutes. Stir in sugar.

Whisk whole wheat flour, all-purpose flour, oatmeal, and salt together in a mixing bowl; make a well in the center. Pour milk mixture into well and stir until dough has pulled together. Knead dough on a lightly floured surface until smooth and elastic, about 8 minutes.

Place dough in a large lightly oiled bowl and turn to coat. 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. Meanwhile, grease two 8x4-inch loaf pans and set aside.

Punch down dough to deflate and turn onto a lightly floured surface; divide into two equal pieces. Shape into balls and let rest for 10 minutes. Form dough into loaves and place into prepared pans. Cover loaves with a damp cloth and let rise until doubled in volume, about 40 minutes.

Preheat the oven to 350 degrees F (175 degrees C).