Finely chop the olives until paste-like. Whisk in half of the olive oil and 1 Tbs. of the herbs.
Combine the remaining herbs and oil with the garlic in a small bowl.
Rub the herb and garlic marinade all over the lamb and wrap it well in plastic wrap. Let sit at room temperature for 1 hour.
When ready to grill, prepare a medium (375°F) charcoal or gas grill fire. Season the lamb generously with salt and pepper. Grill the lamb, flipping once, until the thickest part is 120°F for medium rare, 15 to 18 minutes.
Transfer the lamb to a cutting board, squeeze the lemon half over it, tent with foil, and let rest for 10 minutes. Thinly slice across the grain, drizzle any accumulated juice over it, and serve with the olive purée.
1 cup pitted black olives
1/2 cup extra-virgin olive oil
1/2 cup finely chopped fresh herbs, such as a mix of parsley, rosemary, and thyme
3 large cloves garlic, minced and mashed to a paste with a pinch of salt
4 to 4-1/2 lb. butterflied leg of lamb
1/2 large lemon, seeded
salt and freshly ground black pepper