Preheat your oven to 375°F (190°C) and line a baking sheet with parchment paper.
Using a sharp knife, carefully cut through the center of each bagel horizontally, creating a pocket while keeping the sides attached.
Gently scoop out the soft interior of each bagel half with a small spoon or scoop, leaving a sturdy shell for the filling.
In a mixing bowl, whisk together the eggs, milk, chopped green onion, and a pinch of salt and pepper until well combined and slightly frothy.
Fold in the shredded cheddar cheese into the egg mixture, distributing evenly for a cheesy filling.
Pour the egg mixture into each hollowed-out bagel shell, filling them about three-quarters full.
Arrange the filled bagels on the prepared baking sheet and bake in the preheated oven for about 12-15 minutes, or until the filling is puffed and just set, and the bagels are lightly toasted.
Remove from the oven and let cool slightly before serving. The bagels should be crispy on the outside with a creamy, cheesy egg center inside.