Preheat the oven to 400°F and line a baking sheet with parchment paper.

Cook the breakfast sausage in a skillet over medium heat until browned and fully cooked, breaking it into small crumbles. Transfer to a plate and set aside.

In a medium bowl, whisk together the eggs, milk, salt, and black pepper until smooth.

Wipe out the skillet if needed and melt the butter over medium-low heat. Add the egg mixture and cook, stirring often, until the eggs are softly scrambled and just set. Remove from the heat.

Arrange the bagel halves cut-side up on the prepared baking sheet. For a crisper base, pre-toast them in the oven for 2 to 3 minutes if desired.

Divide the scrambled eggs evenly over the bagel halves, then top with the cooked sausage.

Sprinkle the shredded mozzarella evenly over each bagel half.

Bake for 8 to 10 minutes, until the cheese is melted and bubbly and the bagel edges are toasted.

Let cool briefly, garnish with chopped green onions if using, and serve warm.