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Use a swivel blade peeler to shave the asparagus into ribbons. Put in a bowl and season with salt and pepper, then dress with the lemon juice and olive oil and set aside.
Mix the parsley, capers, gherkin and lemon zest into the mayonnaise in a small bowl, then gently fold through the crabmeat. Toast or griddle the sourdough and top with the crab mixture. Toss the rocket in with the asparagus and top each crab toast with a tangle of the asparagus salad. Season with flaky sea salt and serve.