1. For the tomato vinaigrette, whisk together the tomato paste and ½ cup of the olive oil in a heavy-bottomed skillet or saucepan until well combined. Warm the mixture over low heat, stirring as needed to prevent sticking, for about 15 minutes, to infuse the oil with the tomato flavor. Keep the temperature low to prevent browning. Remove from the heat and let cool. Whisk together the garlic, red wine vinegar, Parmesan, salt, pepper and oregano in a medium bowl. Whisk this into the cooled tomato oil. Gradually whisk in the remaining ½ cup of olive oil. The texture will be quite thick. If you prefer a thinner consistency, whisk in a little water. Allow the vinaigrette to sit for at least an hour to allow the flavors to meld. (Makes about 1 ½ cups; will keep refrigerated for 1 week.)
2. To make the carpaccio, soak zucchini in cold water for 20 minutes to hydrate and firm up. Slice the zucchini into ¼-inch-thick disks and arrange evenly on a large plate or platter. Sprinkle with salt and pepper. Add the mozzarella (if using). Spoon or drizzle about ¼ cup of the vinaigrette over the zucchini and cheese, adding more as necessary. The ingredients should be very well dressed but not submerged. Scatter the shaved Parmesan and torn basil leaves over the platter. Drizzle with olive oil and season with salt and pepper. Serve with crusty bread to mop up all of the delicious vinaigrette.
Note: A fruity vinegar works best in this recipe. If you don’t have one, add a pinch of sugar or spoonful of honey.