1 large egg, lightly beaten, plus 2 large eggs beaten with 1/4 cup water
1 teaspoon salt
1/4 teaspoon freshly ground pepper
1/2 cup all-purpose flour
2 cups Japanese panko bread crumbs (see Note) or coarse dry bread crumbs
Canola oil, for frying
Preheat the oven to 400°. Bake the potatoes for about 45 minutes, or until tender. Scoop the flesh into a large bowl and mash, then let cool.
Meanwhile, in a saucepan, bring the barley and water to a boil. Cover and cook over low heat until tender, about 20 minutes. Drain the barley and spread it out on a plate to cool.
Add the cooled barley and the 1 beaten egg to the potatoes and mix well. Add the salt and pepper and shape the mixture into eight 1/2-inch-thick cakes. Put the flour, the 2 eggs beaten with water and the panko in separate bowls. Dip each cake into the flour, shaking off the excess. Dip the cakes in the eggs, then coat thoroughly with panko crumbs. Redip the cakes in the flour, eggs and panko and set them on a large plate.
Preheat the oven to 300°. Heat 1/8 inch of oil in a large skillet. Add 4 of the potato cakes at a time and cook over moderate heat until well browned, about 3 minutes per side. Lower the heat if necessary and add more oil to the pan as needed. Transfer the cakes to a plate and keep warm in the oven until serving.
The potato cakes can be prepared through Step 3 and refrigerated for up to 2 days.
Japanese panko bread crumbs are unseasoned and coarse textured. They're popular with chefs because they absorb less fat than regular bread crumbs and stay crisper after frying. They're available at Asian markets.