Put cornmeal in a large bowl with salt and cheese. Put milk in a small saucepan over medium-low heat until steam rises, then add butter and stir until melted. Remove from heat and stir into cornmeal mixture until a thick batter is formed. Fold in the corn kernels, scallion, cilantro and chile, if using.
Let batter rest until it thickens into a soft dough, about 15 minutes. Gently form 3-inch to 4-inch balls from mixture and flatten with palm of your hand to a 1/2-inch-thick disk. (You can cover and refrigerate disks for a few hours if you like.)
Heat oil in a large skillet and cook arepas, working in batches, until golden brown, about 5 minutes, then flip and cook for another 3 minutes on other side. When all arepas are cooked and cool enough to handle, carefully slice them through the middle. If desired, serve with butter or stuff with beans, vegetables or sour cream.