Assemble salad: Cut the burrata into equal segments and place the segments cut side up on each of six plates. Spoon 1 tablespoon of pesto over each portion of cheese. Use the scissors to cut the tomato vines as necessary, dividing the tomatoes into six clusters, and place one cluster atop each serving of cheese. Garnish each plate with fresh basil and sprinkle with sea salt before serving.
Add the pine nuts, grated garlic, and salt to a mortar; gently pound with pestle until a course paste forms. Chop the basil leaves, add them to the mortar, and continue to pound and grind until the basil is integrated with the pine nut mixture. Stir in the lemon zest and slowly drizzle in the olive oil while gently pounding the mixture. Add the lemon juice and cheese. Taste for seasoning and add more salt of lemon juice if desire.
Tomatoes: Adjust the oven rack to the centre position and preheat the oven to 300 F. Keeping the tomatoes attached to the vine, gently place them on a wire rack fitted in a rimmed baking sheet. Drizzle the olive oil on the tomatoes, then sprinkle with salt. Roast until the skins are shriveled but the tomatoes are still plump, about 1½ hours. Remove from the oven and allow to cool to room temperature.