In a large bowl, combine potato puree, broccoli puree, cheese, egg, 3 Tbs. of panko, salt, and garlic powder; mix well to combine.
Place the remaining cup of panko in a shallow dish; set aside.
Shape rounded tablespoonfuls of the potato mixture into balls, then flatten slightly to form nuggets. Dip each nugget in panko, using your hands to gently pat on the coating.
Heat 1 Tbs. oil in a large nonstick skillet over medium-high heat. Add half the nuggets; cook 3 to 4 minutes on each side or until golden brown. Remove from pan. Repeat procedure with remaining oil and nuggets.