Preheat grill to medium heat.
Drizzle olive oil over the broccoli halves. Season with salt and pepper.
In a small skillet, add about ½ tablespoon of olive oil, the breadcrumbs, and season with salt and pepper. Toast in the skillet until golden and crispy. Remove from the pan and set aside.
In a large bowl, whisk together mayo, Dijon, garlic, lemon juice, Worcestershire, anchovy paste, and Parmesan. Cover and chill.
Grill the broccoli for 3-4 minutes per side, until tender and lightly charred. Transfer to a serving platter.
Drizzle on some of the dressing. Top with shaved Parmesan and the toasted breadcrumbs. Serve with lemon wedges.