Tip all of the ingredients for the pesto, except the oil and lemon juice into a food processor until a rough crumb forms. Pour in the oil and lemon juice and pulse again to combine. Transfer to an airtight container and store in the fridge until ready to use (it will keep covered for up to a week).
Unroll the puff pastry onto a flat baking sheet and prick all over with the prongs of a fork. Score the pastry 1cm from each edge to create a border, then brush the border and centre with the beaten egg mixture. Spread ½ the pesto over the top of the tart (saving the rest for another meal), and arrange the asparagus spears over the pesto, so that they’re evenly spaced. Brush the asparagus with oil, and pour any remaining beaten egg between the asparagus.
Slide the tart into the oven to cook for 22-25 minutes until puffed up and golden. Remove from the oven and allow to cool for a few minutes before slicing and serving. This makes an excellent picnic tart, or delicious supper served alongside dressed salad and new potatoes.
Tip: If you are cooking for anyone with a nut allergy, swap out the nuts for toasted pumpkin or sunflower seeds instead.