Position a rack in the center of the oven and preheat to 400 degrees . In a large roasting pan, stir together the EVOO and anchovies. Add the broccoli, chestnuts, 1/2 tsp. salt and 1/4 tsp. pepper; toss to coat. Roast, stirring once, just until the broccoli is crisp-tender, about 15 minutes.
Meanwhile, in a small bowl, combine the parsley, garlic and lemon zest. Stir the parsley mixture into the broccoli mixture and spoon into a serving dish.