Preheat oven to 350 degrees F. Place pistachios on a baking pan and roast for 8-10 minutes, stirring once or twice. Remove from oven, transfer to a plate and allow to cool.
Raise oven temperature to 425 degrees F. Line a baking sheet with parchment paper or foil. Toss Broccolini with olive oil on baking sheet, making sure that the Broccolini heads are oiled, then season with ¼ teaspoon each salt and black pepper. Spread the Broccolini so that they don’t overlap and roast until the florets are charred a bit, 12-17 minutes. Remove from the oven and allow to cool. When they are cool, cut the stalks in half crosswise.
Place the sliced shallot in a small bowl of ice water while you prepare the other ingredients. Drain well and pat dry before adding to salad.
While the Broccolini bakes, tear the lettuce leaves and slice the kale.
Meanwhile chop anchovy fillets into very small pieces, then mash them with a fork. Whisk together the mayonnaise, vinegar, mustard, and garlic in a small bowl. Season liberally with black pepper and stir in anchovy paste. Mix well.
Place the Broccolini, lettuce, and kale together in a large serving bowl. Add the dressing and toss. Sprinkle on the shallots, chickpeas, and Parmesan. Toss again. Sprinkle with pistachios and serve.