Arugula Pesto: Pulse arugula, garlic, cheese, walnuts until smooth. Then gradually pour in the olive oil until well emulsified. Add a generous squeeze of lemon juice. Season pesto with salt and pepper to taste. You will get approximately 1 cup of pesto but you only need 6-8 tablespoons for this tart. Keep the rest refrigerated for up to 3 days or freeze it if desired.
Preheat the oven to 395 F and line a baking sheet with parchment paper.
Puff pastry: Preheat the oven to 395 F and line a baking sheet with parchment paper. According to the packaging directions, if required, roll out the pastry slightly on a lightly floured surface. Transfer to the lined baking sheet and prick the pastry all over with a fork. Lay another sheet of parchment paper on the puff pastry and place a second baking sheet on top, bake for 15-20 minutes, until golden. Remove the top sheet and discard the paper, then bake for 5 minutes more or until crisp and dry. Cool slightly.
Mushrooms: Sear the mushrooms (Alba & Brown Clamshell, Forest Nameko, Velvet Pioppini varieties) in a pan on medium-high heat in some oil for 4-5 minutes until nicely browned on all sides. Season well with some sea salt and freshly ground black pepper. Transfer the mushrooms to a plate, lined with paper towel to absorb the extra oil. Cut Trumpet Royal mushrooms in 3 slices lengthwise each and sear them for 2 minutes on each side until browned.
Assemble: Spread arugula pesto over the pastry, leaving a ½ inch border. Scatter some baby arugula and/or baby kale leaves on top. Tear the prosciutto into bite size pieces and lay over the pastry. Scatter seared mushrooms in between prosciutto and sprinkle generously with the remaining cheese on top. Lightly beat egg yolk and use it to brush the pastry border. Season with a pinch of red pepper fakes, bake for 5-8 minutes until the cheese melts.