Place the feta in the bowl of a food processor fitted with the blade attachment. Pulse into small crumbs, 6 to 8 pulses. Add the yogurt and blend, scraping down the sides of the bowl as needed, until the mixture is whipped and very smooth, for 3 to 4 minutes.
Add the lemon zest, garlic, and a few generous grinds of black pepper, and blend for 10 seconds more. Transfer the whipped feta to a serving bowl and drizzle with the pesto before serving with crackers, bread, or vegetables.