The afternoon before, call over 4 pals for a prep party! Ask each to bring a large pot and large bowl, and maybe an extra blender. For each batch (5 total): Chop 3 onions and 6 garlic cloves. Cook in 6 tbsp. oil in a large pot over medium heat, stirring often, until translucent, about 7 minutes.
As a group, prep tomatoes, peppers, and cucumbers.
Take turns at the blender(s): Purée each batch of cooked onions and garlic with one- fifth of tomatoes and 5 to 6 tbsp. tomato paste, if using*, until very smooth, about 2 minutes per batch. Pour each into a large bowl.
Finish the soup: For each batch, blend one-fifth of peppers, one-fifth of cucumbers, 3⁄4 cup oil, 31/2 to 4 tbsp. vinegar, 11/2 tsp. white pepper, and 1 tbsp. fine sea salt until very smooth, about 2 minutes. Whisk into bowl of tomato purée. Strain, then season with more pepper, salt, or vinegar. Chill until very cold, at least 4 hours or overnight.
Divide and conquer: Cook eggs and shrimp (at right); slice onions; chill.
Four hours before serving, make toasts (no earlier or they’ll get stale): Preheat a grill to 350°. Grill bread slices in batches until lightly charred, 3-5 minutes, and transfer to platters or baking sheets in a single layer. (Alternatively, broil in oven 2 to 3 minutes to toast.) Let toasts cool a little, then rub with garlic cloves. Set aside, uncovered at room temperature, until serving. Shortly before serving, drizzle with olive oil.
Two hours before, plate ham slices; peel, chop, and re-chill eggs.
One hour before, bring those friends back to help you serve. Fill large mason jars with soup. Set up 3 stations with soup (nestle into ice-filled trays), toppings, accompaniments, and serving ware.
*If your tomatoes aren’t perfectly ripe, add 6 tbsp. good-quality tomato paste to each batch of soup to boost the flavor.