Whisk the egg with a fork in a small bowl. In a separate bowl, stir together the flour and pepper. Heat the oil in a heavy skillet or deep fryer to 375 degrees F (190 degrees C). Dip each onion ring into the egg and then coat with flour. Add them to the hot oil a few at a time. Fry until golden brown, turning once if needed, it should take about 3 minutes. Set them onto a paper towel-lined plate to drain and cool. Chop into 1/2 inch pieces.
Preheat the oven to 350 degrees F (175 degrees C).
In a medium bowl, stir together the green beans, cream of mushroom soup, and milk. Stir in about half of the onions. Transfer to a casserole dish.
Bake in the preheated oven until heated through and bubbly, about 25 minutes. Remove from the oven and sprinkle the cheese and remaining onions on top. Bake until browned on the top, about 10 more minutes. Remove from the oven and garnish with a sprinkle of paprika and black pepper.