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) 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 and let rest for 10 minutes. Thinly slice across the grain and serve with the olive purée.
1 cup pitted black olives, such as Kalamata
1/2 cup extra-virgin olive oil
1/2 cup finely chopped fresh herbs, such as a mix of parsley, rosemary, and thyme
4 large cloves garlic, minced and mashed to a paste with a pinch of salt
4 to 4-1/2 lb. butterflied leg of lamb
Kosher salt and freshly ground black pepper
1/2 large lemon, seeded