How to Make It
Whisk together vinegar and tomato paste in a large bowl until smooth. Stir in tomatoes, bread, garlic, salt, and pepper. Chill mixture, covered, until bread is soggy, about 15 minutes.
Ladle half of the tomato-bread mixture into a blender and add ¾ cup water. Process until mostly smooth, about 1 minute. Add remaining tomato-bread mixture and process, stopping to scrape down sides of bowl as needed, until smooth, about 2 minutes. With motor running, drizzle in oil until combined, about 1 minute. Chill soup, covered, for at least 1 hour and up to 1 day.
To serve, ladle chilled soup into bowls and top with ham and chile. Using a Microplane or the smallest holes of a box grater, grate eggs over tops of bowls. Garnish with several grinds of pepper.