Mix mayonnaise, ketchup, 1 teaspoon of the sliced green onions and 2 teaspoons of the parsley in a bowl. Set aside.
Cut the chicken breast in half down the center. Cut each half into four thin long slices.
Peel the avocado, remove the pit and slice each half lengthwise into four slices.
Place two slices of chicken on each piece of toast. Spread half the reserved dressing over the chicken. Lay the avocado slices over the dressing and cover with remaining dressing. Place a piece of cheddar on top of each sandwich (the cheese should be large enough to cover the toast). Sprinkle lightly with the paprika and the remaining green onion slices.
Preheat the broiler.
Place the sandwiches in a broiling pan and broil 3 inches from the heat until the cheese melts. Remove the pan from the broiler and sprinkle the sandwiches with the remaining 1 teaspoon of parsley. Serve immediately.