Heat the olive oil in a small skillet until it shimmers. Add the garlic. When you can see the first signs of browning, add the thyme and remove the pan from the heat. Spoon the entire contents of the pan into a large bowl to cool.
In two or three batches (depending on how big your food processor is), combine all the remaining Ingredients in a food processor. Process until smooth. (A few small solid grains are okay.)Check seasonings and add more salt and pepper, if needed.
Combine all the batches in the bowl with the olive oil and garlic. Mix with a rubber spatula until well blended.
Garnish with extra-virgin olive oil, olive, capers, caper berries, and fresh chopped parsley. Serve with crusty bread as an appetizer.