Process sorrel, Parmesan, pine nuts, lemon juice, and 1/2 teaspoon salt in a food processor until coarsely ground. With machine running, add oil in a slow, steady stream until mixture is emulsified. (Pesto may be refrigerated in an airtight container for up to 3 days.)
Spread 2 tablespoons pesto on each flatbread, and top with 1/2 cup edible weeds. Drizzle with oil, and sprinkle with salt. Serve immediately.