Preheat the oven to 500 degrees F.
To a medium skillet over medium heat, add the olive oil and heat until it begins to shimmer. Then add the sausage, cooking and breaking it up into crumbles with a wooden spoon until it begins to brown, about 5 minutes. Transfer the sausage to a medium bowl with a slotted spoon and set aside to cool slightly.
Add the garlic to the skillet. Stir to combine, being careful not to burn it (reduce the heat if necessary). When the garlic is fragrant and just translucent, add the broccolini and toss to coat in the garlic and olive oil. Cook until just tender, about 7 minutes. Set aside to cool slightly.
Meanwhile, to the medium bowl with the cooked sausage, add the mozzarella, ricotta, Parmesan, lemon zest and black and crushed red pepper to taste. Stir to combine.
Divide the pizza dough into four equal parts and roll into balls. On a lightly floured work surface, flatten them into circles, about 1/4 inch thick. Spread them each with one-quarter of the cheese mixture, sausage and broccolini, leaving a 1-inch border around the edges. Fold them in half to make a calzone shape, pinch the edges to seal and then transfer the pizza pockets to a pizza peel, pizza stone or baking sheet, 2 inches apart. Bake until lightly browned on top, 10 minutes. Let cool slightly and serve with warm marinara sauce.