Snap the woody ends off the asparagus. Cook the asparagus and beans in a frying pan of simmering salted water for 5 minutes or until tender.
Using a vegetable peeler, slice the parmesan into thin shavings over a sheet of paper. Allow for 6 or 8 shavings per person.
In a large bowl, whisk the walnut oil, extra virgin olive oil, white wine vinegar, mustard, garlic, sea salt and pepper together.
Drain the asparagus and broad beans, shake dry, and toss in the dressing while still hot. Add the tarragon or mint, and turn on to four plates. Gently scatter the parmesan shavings on top and serve.