Preheat the oven to 425°F (220°F). Set a rack in the middle of the oven. Line a rimmed baking sheet with aluminum foil and place it in the oven while it heats.
In a bowl, toss together the Brussels sprouts, olive oil, salt, and a pinch of black pepper. Arrange the brussels sprouts on the hot baking sheet in a single layer, cut-side down.
Roast for 15 minutes. Flip the brussels sprouts and sprinkle with two-thirds of the Parmesan cheese. Continue roasting until the Brussels sprouts are deeply golden, crispy, and tender, 12 to 15 minutes more.
Meanwhile, in a medium bowl, combine the mayonnaise, lemon juice, anchovy paste, Worcestershire sauce, mustard, the remaining Parmesan cheese, and a generous pinch of salt and pepper. Whisk to combine, then scoop the dip into a small serving bowl.
Transfer the Brussels sprouts and surrounding crispy Parmesan bits to a platter. Serve with the Caesar dip.
