Place the eggs in a small pan of water and boil for 5 minutes, then drain and plunge into a bowl of iced water. Set aside for 10 minutes as this will make them much easier to peel. Preheat the grill to high. Cut the bagels in half and arrange on a grill rack. Cook for 1-2 minutes until toasted. Spread with a little butter. Place the toasted bagel halves on warmed plates and ruffle the smoked salmon slices on top. Add half a tablespoon of crème fraîche.
Shell the eggs and cut each one in half, then arrange on top of the smoked salmon and crème fraîche. Season lightly and sprinkle with the chives to serve.