Prepare a charcoal or gas grill for direct cooking over medium-high heat. Trim broccoli and cut lengthwise into 1-inch-thick steaks. Toss with olive oil and salt.

Grill broccoli for 12 to 16 minutes or until charred, turning halfway through. Remove from grill; cool. Meanwhile, combine ¼ cup olive oil, fresh parsley, sesame oil, tahini paste, garlic, and lemon juice in a food processor. Cover and pulse just until combined. Add broccoli and pulse until smooth. Transfer mixture to a serving bowl. If desired, drizzle with additional olive oil and sprinkle with toasted sesame seeds, if desired.