Rinse and drain the capers and olives to remove excess salt. Note: This step is optional, but I find it helps mellow the briny flavor. For best results, I recommend soaking them in water for 10–15 minutes before rinsing and draining.
In a food processor, combine the olives, capers, anchovy fillets (if using), garlic, and thyme leaves.
Add the lemon juice and two tablespoons of olive oil.
Blend in short pulses until you reach your preferred consistency. Tapenade can be chunky or smooth, depending on your preference. I usually pulse 2 to 3 times since I like a chunkier texture.
Season with freshly ground black pepper.
Transfer to a bowl, drizzle with the remaining tablespoon of olive oil, and serve with crusty bread, crackers, or fresh vegetables.