Cut the bagels in half (horizontally). Put the bagel halves in a baking tray cut-side facing upwards. Spread some pesto on the bottom halves of the bagels, then cover with thin slices of cheese.
Lightly beat the eggs with a pinch of salt. You can add a drop of milk for a lighter texture.
Turn the grill on to medium. When it’s hot, put the bagels underneath the grill and cook for a minute or so, until the cheese has melted and the top sides of the bagels look golden brown. Remove from the oven.
Meanwhile, heat some butter or oil and add the eggs, then keep stirring for a minute or so. Add plenty of freshly ground pepper. Divide the scrambled egg between the cheesy sides of the bagels, then put a slice of Parma ham on it. Scatter some rocket salad on top and cover with the top half of the bagel.
Serve immediately.