Place anchovies, capers, caper brine, and garlic in food processor or blender and turn the machine on. Stream in the olive oil and form a thick dressing. Scrape the dressing into a small serving cup or bowl and return the processor bowl or the blender to its base.
Black Olive and Parmigiano Sauce:
Place the black olives, vinegar, and thyme in processor or blender. Turn machine on and stream in the olive oil. When dressing forms, stop machine and add cheese. Pulse-process cheese into dressing then transfer dressing into a second cup or small bowl.
Garden Tomato and Green Onion Sauce:
In a small mixing bowl, combine tomatoes, scallions, and parsley. Drizzle the raw sauce with the olive oil and season it with salt and pepper, to your taste.
Place the in a small pot. Cover eggs with water and bring water to a boil. Cover pot and remove from heat. Let eggs stand 10 minutes.
Drain eggs and crack shells by rattling cooked eggs inside empty covered pot. Peel and cool eggs under cold running water.
Arrange the vegetables in a flower pot, building high and stuffing the pot tightly. Nest whole eggs and whole tomatoes among the vegetables. Serve your vegetable masterpiece with sauces to pass at table.