Line colander with paper towels. Place potatoes and shallots in colander, and let stand for 15 minutes. Squeeze out excess moisture.
Place bread in a food processor; pulse 10 times or until coarse crumbs measure 1/4 cup. Lightly spoon flour into a dry measuring cup; level with a knife. Combine potato mixture, breadcrumbs, flour, sugar, salt, pepper, and egg in a bowl. Divide mixture into 8 portions. Pat each portion into a 1/4-inch-thick patty.
Melt 1 1/2 teaspoons butter in a large nonstick skillet over medium-high heat. Add 1 1/2 teaspoons oil to pan. Add 4 patties to pan; cook 3 minutes on each side or until golden brown. Repeat procedure with remaining butter, oil, and patties.