Heat 1 tablespoon oil in a skillet over medium-high heat. Saute onion in hot oil until golden and soft, 7 to 12 minutes.
Lightly oil a large skillet and place over medium heat. Place 1 tortilla in the skillet and top tortilla with 1/4 of the refried beans, leaving a space in the center of the tortilla. Crack 1 egg into the center of the tortilla. Sprinkle 1/4 of the onion, 1/4 of the Cheddar cheese, and 1/4 of the chopped green chilies over egg and beans. Place 1 tortilla over the top. Cook, turning once, until egg is set and each tortilla is browned and crispy, 3 to 6 minutes per side.
Transferred finished "pizza" to a plate and repeat process with remaining tortillas, beans, eggs, onion, cheese, and chilies.