Bring milk to a simmer in a small saucepan, then remove from heat and stir in butter.
Combine arepa flour, salt, sugar, and mozzarella in a large bowl.
Add hot milk and stir until combined.
Let mixture stand until milk is absorbed enough for a soft dough to form, 1 to 2 minutes (dough will continue to stiffen).
Form dough into 12 balls (about 2 inches in diameter) and flatten between palms into 3 1/2- to 4-inch patties (about 1/3 inch thick).
Heat ‚ tablespoon oil in a large nonstick skillet over moderately low heat until hot, then cook 3 or 4 arepas until cooked through and golden in patches, 6 to 8 minutes on each side. Make more arepas in same manner, adding oil as needed.