Preheat the Oven:
Preheat your oven to 375°F (190°C) and line a large baking sheet with parchment paper.
Prepare the Bagels:
Using a small biscuit cutter or knife, enlarge the center holes of the bagel halves to about 3 inches in diameter. This will make room for the egg.
Butter the Bagels:
Brush the cut sides of each bagel half with melted butter, then place them cut-side down on the lined baking sheet. You can also add the bagel cut-outs to the sheet to toast them, if desired.
Assemble the Filling:
Place a slice of cheese inside each bagel hole, followed by a slice of prosciutto. Tuck the cheese and prosciutto into the bagel hole to create a little nest.
Add the Eggs:
Crack an egg into the center of each prepared bagel, letting the cheese and prosciutto form a barrier to hold the egg in place. Season the eggs with salt and pepper.
Bake the Bagels:
Transfer the baking sheet to the preheated oven and bake for 10-15 minutes, depending on how you prefer your eggs. Around 12 minutes, the whites should be set with a slightly runny yolk.
Serve:
Once the eggs are cooked to your liking, remove the bagels from the oven and garnish with freshly chopped chives. Serve while warm and enjoy!