Place milk, water, butter, salt, sugar, onion powder, 3 tablespoons dried onion, potato flakes, flour, and yeast into the pan of a bread machine in that order, or follow the order recommended by the manufacturer if different. Select the Dough cycle and press Start.

When the cycle has completed, remove dough from the machine and knead on a lightly floured surface. Cut into 8 equal pieces and form into balls. Gently flatten the balls until they are 4 inches in diameter. If they keep shrinking back, just let them relax for a minute before flattening. Place on a baking sheet and cover loosely with a towel. Set in a draft-free place to rise until doubled in size, about 40 minutes.

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

Whisk together egg white and water in a cup. Brush over the tops of the risen rolls and sprinkle with 1/4 cup minced onion.

Bake in the preheated oven until golden brown, 15 to 20 minutes.

Cool completely then slice in half horizontally before using.