In a large stockpot bring the water to a boil, add two good pinches of sea salt and then add pasta. Cook for 2 minutes short of suggested time. Add the green beans the last 3 minutes of the pasta cooking time. Reserve 1 cup of pasta water. Drain pasta and green beans.
While the pasta is cooking, heat a 12 inch sauté pan over medium high heat. Add two tablespoons of olive oil. Add the eggplant and sauté, stirring until tender about 4-5 minutes. Add garlic and sauté for 30 seconds. Add 2 cups of the tomatoes and sauté for about 3 minutes. (Sprinkle with Piment d' Espelette of using) Add the wine and cook, stirring until the wine is almost entirely evaporated.
Add the pasta, green beans, and remaining tomatoes. Stir. Add the butter and Parmigiano-Reggiano cheese, stir to combine, Add some of the pasta water if mixture is too dry.
To serve; put pasta in a large flat pasta bowl and garnish with mozzarella pieces, and top with mint/basil and scallions. Drizzle with olive oil and sprinkle with salt flakes and black pepper.