Mix flour, baking powder, salt, and pepper together in a separate medium bowl. Add milk and egg; stir until batter is smooth. Let stand for 20 minutes.

Meanwhile, heat oil in a large pot or electric skillet to 350 degrees F (175 degrees C). Drain and pat dry potatoes. Fry in hot oil until tender, about 5 minutes; drain on paper towels.

Dredge cod in batter, one piece at a time, and place in hot oil. Fry fish in batches until golden brown on all sides; drain on paper towels. Monitor oil temperature, increasing the heat as needed to maintain 350 degrees F (175 degrees C).

Fry potatoes again in hot oil until crisp, 1 to 2 minutes; drain on paper towels.