Remove 1 puff pastry sheet from the freezer and let it defrost on the counter for 40 minutes on top of parchment paper. Meanwhile, preheat the oven to 400º F and make the filling.
Drain and rinse the beans, and add them to the work bowl of a food processor. Add of the remaining ingredients except the soymilk (salt through lemon juice). Pulse the food processor, stopping ever few pulses to scrape down the sides and pulse again. You want to break up the beans and the cashews. Once it’s as smooth as you can get it, dribble in the soymilk while the machine is running, as this should help further smooth the mixture.
After 40 minutes of defrosting, carefully unwrap the puff pastry on top of your parchment paper.
Roll the puff pastry out slightly into a rectangle. Then add the white bean pesto mixture, leaving a 1 to 1 1/2 inch border on all sides.
Trim your asparagus so that it fits the tart width-wise. Press the spears into the white bean pesto, alternating their direction so that there are tips and bases next to each other; this one everyone is guaranteed every part of the asparagus after cutting.
With a sharp knife, trim all the edges to they are neat and square. Trim away as little as possible.
Brush the spears with olive oil and sprinkle lightly with salt and pepper.
Using the parchment paper, slide the tart onto a baking sheet (don’t remove the tart from the paper). Bake at 400ºF for 25-30 minutes until the tart is puffed and golden brown and the spears are tender. Serve while just warm.
You can reheat this tart in the oven for a few minutes if you have leftovers, but it’s also good at room temperature. Fresh basil is a nice garnish. If you’ve never worked with puff pastry before, this is a great recipe to get your feet wet. It really is easy to use, just as delicious as phyllo dough with none of the stress. And it’s fun to watch it puff!