To make in a blender: Place tomatoes, ½ cup basil leaves, 1 Tbsp white wine vinegar, 2 tsp Kosher salt, and ¼ tsp black pepper in a blender, then process until puréed. With the motor running on low, drizzle in ⅓-½ cup olive oil. Continue blending until very smooth, then taste and adjust seasoning as needed. If not using right away, cover the soup with plastic wrap and store in refrigerator until serving.
To make in a food processor. Place tomatoes in the bowl of a food processor fitted with blade attachment. Add ½ cup basil leaves, 1 Tbsp white wine vinegar, 2 tsp Kosher salt, and ¼ tsp black pepper, then process until smooth. With the motor running, drizzle in ⅓ - ½ cup olive oil. Because a food processor won't purée as well as a blender, you have the option to strain the mixture so that it's smooth.
Strain the gazpacho. Place a fine mesh sieve under a large bowl, then pour the tomato mixture into the sieve. Use a rubber spatula to push the sauce through, leaving behind only the pulp and small bits of basil (discard). If not using right away, cover the soup with plastic wrap and store in refrigerator until serving.
Optional: Make the basil oil. Rinse the bowl of the blender or food processor with water, then shake any excess liquid out. To make in a food processor: add remaining 1 cup basil leaves to the bowl, then process until finely chopped. With the processor running, drizzle in ¼ cup olive oil. Once again, place your fine mesh sieve on top of a bowl, then strain basil oil. Set aside. To make in a blender, combine 1 cup basil and ¼ cup olive oil in the blender bowl, then process until puréed. Strain the mixture, if desired.
Assemble, then serve. Divide tomato gazpacho into shallow bowls. Tear fresh mozzarella with hands into small, bite-size pieces, then place directly on top of the soup. Use a spoon to drizzle a small amount of basil oil (or extra virgin olive oil) over the top of the dish. Finish with flaky sea salt, freshly ground black pepper, and additional fresh basil leaves.