In a medium bowl, use your hands to mix together the masarepa, salt, water, and butter.
Knead until a smooth dough comes together, about 3 minutes. Moisten your hands as needed with water to prevent sticking. Cover and allow to rest at room temperature for 5 minutes.
Divide the dough into 6 equal pieces. Gently flatten each piece into a circle 1/3-1/2 inch (8-13 millimeters) thick and about 5 inches (13 centimeters) wide.
Grease a large skillet with butter and place over medium heat. Place the arepas in the heated skillet, in batches to prevent overcrowding, and cook until golden, about 5 minutes per side.
Transfer to a towel lined plate and repeat with remaining arepas. If the butter starts to darken too much, add a drizzle of oil.
Split each arepa open through the middle, leaving about 1/2 inch (1.25 centimeters) of one edge attached. Fill each arepa with a handful of the shredded cheese.
Place back onto the hot skillet and cook on each side until heated through and cheese is melted.