First, for cooking the pasta shells, make sure you salt the water!
Preheat oven to 350ºF.
Heat olive oil in a medium sized pan on medium-high heat. Add the eggplant and a good pinch of salt. After the eggplant have browned slightly (3-4 minutes), add the broccolini. After 2 minutes, add the garlic and cook another 2 minutes. Take the pan off the heat and add the pasta. Gently stir, and then add the mozzarella and goat cheese over the top. Break basil leaves over the top and bake in the oven for about 7 minutes, or until the cheese is just melted.