Soak dried apricots and pistachios in boiling water for 10 minutes, then drain.
Chop garlic into large pieces.
Add apricots, pistachios, garlic, green olives, basil, lemon juice, olive oil, and pepper to a food processor. Blend until desired texture, scraping sides as needed. Add salt to taste. If too thick to blend, add more olive oil.
Chill for a couple hours before serving. Store leftovers in an airtight container in the fridge for up to 5 days, with a thin layer of olive oil on top.