Put the eggplant in a colander and sprinkle with 2 teaspoons kosher salt. Toss the eggplant and set the colander in the sink to drain for 10 minutes. The salt will draw out excess moisture and bitterness.)
Meanwhile, combine vinegars, water sugar, and ½ teaspoon kosher salt in a medium heavy pot and bring to a boil over high heat. Carefully add the salted eggplant (the eggplant should be completely submerged in liquid – if it is not, add water to cover) and cook for 1 minute, then remove the pot from the heat and set aside for 30 minutes to lightly pickle the eggplant.
Drain the eggplant pieces and transfer to a medium bowl. Drizzle with 2 tablespoons of the olive oil and toss to coat.
Fill a large pot about 2/3 full with water and bring to a boil over high heat.
Fill a large bowl with ice water. Add the lobsters to the boiling water and cook for 7 minutes, then use tongs to transfer them to the ice bath. Once lobsters are cool, remove the meat from the shells and cut into ¾ inch dice.
Put the tomatoes in a medium bowl and toss with remaining 2 tablespoons olive oil.
To serve place the burrata in the center of a serving plate. Put the lobster meat in a bowl, drizzle with olive oil and a scant bit of lemon juice, season with a pinch of kosher salt, and toss. Artfully arrange lobster, eggplant, and tomatoes around the burrata. Drizzle the basil oil over the plate and garnish with the basil and mâche leaves. Drizzle with extra virgin olive oil and sprinkle with sea salt. Serve immediately.