Step 1 Heat the oven to 400 degrees. Place the grated tomatoes in a large bowl and season with salt and pepper to taste. Taste the tomatoes; if they seem a little bland, add the sugar and vinegar to heighten the flavor. Stir in one-fourth cup of the olive oil.
Step 2 Spread the bread cubes on a rimmed baking sheet and bake until golden and crisp, about 5 to 10 minutes, tossing midway.
Step 3Add the toasted bread cubes and the basil to the tomatoes and toss to mix. Immediately spoon the bread and tomato mixture into martini glasses or large shot glasses, top with the salami and drizzle with olive oil. Serve at once, before the bread gets soggy.