Preheat the oven to 400 degrees F. Line a small baking sheet with aluminum foil and spray generously with nonstick cooking spray.
Place the bagel halves cut-side up on the baking sheet. Use your fingertips to work around the middle of each bagel half, scooping out the excess bread. Be careful to keep the shape and integrity of the bagel; if the bagel tears or cracks, flatten a piece of interior to mend the hole. Sprinkle 1 teaspoon each of the ham, green peppers and red peppers in each bagel half "moat," then sprinkle each with one-quarter of the scallions.
Put the eggs in a container with a pour spout. Whisk in the milk and 1/2 teaspoon each salt and pepper. Slowly pour one-quarter of the egg mixture into each bagel half "moat," making sure the egg does not overflow.
Bake until the eggs are just set and the edges of the bagels are turning golden brown, 12 to 15 minutes. Remove the baking sheet and place a slice of cheese on top of each bagel half. Return to the oven and bake until the cheese is melted and bubbling, about 3 minutes. Serve warm sprinkled with fresh parsley and a dash of hot sauce as desired.
You can flatten the center hole of each bagel half and fill it with the scraps from the interior. This creates a full "bowl" to hold the omelet.
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