Slice the bagels in half horizontally. Carefully widen the hole in the top halves using a knife so the eggs fit comfortably.
Spread butter over the cut sides of the bagels.
Heat a large skillet over low-medium heat and place the bagels cut-side down in the pan.
Crack each egg into a small bowl, then gently pour an egg into the widened hole of each top bagel half. Some egg white may spill out slightly, which is perfectly fine.
Sprinkle the eggs with salt and black pepper.
Cover the pan with a lid or foil and cook for 3–4 minutes until the egg whites are set but the yolks remain soft.
While the eggs cook, scoop the avocado into a bowl. Add lime juice, salt, and pepper, then mash with a fork until creamy.
Place the bottom halves of the bagels on serving plates and spread the avocado mixture evenly over them.
Sprinkle crumbled feta cheese and chopped coriander on top, then drizzle with sweet chili sauce.
Carefully lift the egg-topped bagel halves using a spatula and place them on top of the avocado filling. Serve immediately.