Add milk, one egg, butter, sugar, and salt to the bread pan. Then add white flour, whole wheat flour, and yeast.
Select the dough cycle and press start (takes about 2 hrs).
When the dough cycle is finished, place the dough onto a floured surface and divide into 12 equal portions with a knife.
Shape each portion into a ball and place on a baking sheet.
For the egg wash, crack an egg into a small bowl and mix with one tbsp of water. Whisk with a fork. Use a pastry brush to brush egg wash onto each bun. Let buns rise for 20 minutes.
Bake in 350' preheated oven for 20 minutes (check a little sooner if using a metal baking sheet)