How to Make It
Make the beet hummus: Process beets, parsley, tahini, lemon zest and juice, garlic, salt, and pepper in a food processor until smooth, about 45 seconds, stopping to scrape down sides as needed. With processor running, gradually add oil through food chute until combined and smooth. Set aside.
Make the salad: Stir together shallot, 1 tablespoon vinegar, and 1 tablespoon water in a small bowl. Let stand at room temperature while preparing salad.
Gently toss together tomatoes, cucumbers, feta, olives, oil, salt, pepper, shallots and pickling liquid, and remaining 2 tablespoons vinegar in a large bowl. To serve, spread 3 tablespoons of the beet hummus on each of 4 plates; top each with 1 3/4 cup salad along with any juices in bottom of bowl. Garnish salads with oregano and parsley, and drizzle with oil. Serve with bread.