In a pot of salted boiling water, cook the broccoli until tender, about 3 minutes. With a slotted spoon, remove the broccoli (keep the pot of water). In a food processor, purée with the remaining ingredients until smooth. Use a spatula to scrape down the sides of the food processor as needed. Season with salt and pepper. Set aside.
Place a pizza stone or inverted baking sheet on the middle rack of the oven. Preheat the oven to 450°F (230°C).
In the same pot of salted boiling water, cook the broccoli until tender, about 3 minutes. With a slotted spoon, remove the broccoli and run under cold water. Cook the asparagus in the same water until tender, about 2 minutes. Drain and run under cold water.
Cut the dough into 4 equal pieces. On a lightly floured work surface, roll out one piece of dough into an 11-inch (28 cm) disc. Place on a square of parchment paper. Repeat with the remaining dough.
Cover each pizza with about 3/4 cup (180 ml) of the broccoli cream. Sprinkle with mozzarella. Divide the cooked vegetables and capicollo among the pizzas.
Place 1 pizza at a time (with the parchment paper) on the pizza stone. Cook for 10 minutes or until the dough is cooked through and nicely golden. Immediately garnish with the burratas and kale.