In a large bowl, combine 2 1/2 cups lukewarm water and one-teaspoon kosher salt. Slowly add cornmeal, mixing with hands to combine into dough. If mixture is too dry, add water; if mixture is too wet, add additional cornmeal. On a work surface, knead dough into a solid mass. Let stand a few minutes, knead until smooth. To form arepas, scoop 1 tablespoon. dough and roll into ball. Place ball between two pieces of plastic wrap and flatten into 1/2-inch-thick disk.
In a large skillet, heat 1/2-inch vegetable oil over medium heat. Place arepas in hot oil, without crowding pan. Cook first side about 5 minutes or until golden brown, then cook other side about 3 minutes or until golden brown. Drain arepas on paper towel-lined platter and repeat process to cook remaining arepas. (You should have about 20)
In a small nonstick skillet, melt butter over medium heat. In a small bowl, carefully break eggs while keeping yolks intact. Pour eggs into skillet and cook until whites become solid and edges slightly brown. Gently flip eggs.
To serve, remove skillet from heat and place on table with arepas. Dip arepas in eggs and butter.