Cut off the woody ends of the asparagus and discard them.
Dice the potatoes, chop the asparagus into 2 cm pieces - reserving 4 tips for garnishing.
Soften the onion in the oil, add the potatoes, stir for a couple of minutes, then pour in the stock. Cook for 5-10 minutes, until just tender.
When the potatoes are almost cooked through, add the asparagus pieces, cover and cook for 10 minutes.
Blanch or steam the reserved asparagus tips for a couple of minutes - they should remain crunchy. Rinse immediately with cold water.
Blitz the soup with a hand-held blender or in a liquidizer until smooth, then stir in the lemon juice.
Serve the soup garnished with the blanched asparagus tips and freshly ground pepper on top.