In a medium bowl, whisk together the eggs, milk, salt, and pepper until well combined.
Heat a non-stick skillet over medium heat and add a little butter or oil. Pour in the egg mixture and cook, gently stirring, until the eggs are scrambled and just set, about 3-4 minutes.
While the eggs are cooking, place a slice of cheddar cheese on each half of the toasted bagels.
Once the eggs are done, divide them evenly among the bagel halves. Top each with bacon or sausage, avocado slices, and chopped chives or green onions if desired.
Close the sandwiches with the other half of the bagel and serve warm.