This dish takes advantage of the similarity in size and shape between the giant puffball and Italian eggplant. The individual elements of the dish are cooked separately to insure an attractive presentation, especially considering that the eggplant and the puffball require different cooking times.
Place the whole eggplant in a preheated 350ºF oven for 35 minutes. While it is baking, cook the other ingredients as follows, but don't forget to remove the egglant or it might explode.
Slice the puffball into 1/4" slabs. Heat 1 Tbsp. of olive oil in a large non-stick skillet, medium heat. Place the puffball slices on the pan and dribble some balsamic vinegar on each slice as it cooks. Sautée the puffball slices for about 5 minutes on each side, until slightly brown. The vinegar should evaporate and carmelize. Set them aside.
Slice the shallot (or onion) vertically into paper thin slabs. Don't separate the individual layers, but keep the slabs whole. Spread a bit more olive oil on the skillet, and then place the shallot slabs in the pan. Cook the shallots for a few minutes on their own, until they start to turn golden and translucent. Now add 2 Tbsp. dry sherry (a splash), and allow the sherry to evaporate. Carefully remove the shallots before they burn, and set them aside.
Remove the eggplant from the oven with a towel or oven mitt to avoid burning your hands. Carefully slice the eggplant into disks, about 1/2" thick. You should find that the eggplant is still somewhat rubbery, and will need a bit more cooking. Spread a bit more olive oil on the non-stick skillet. Place the eggplant slices in the skillet, and begin to brown them on one side. Flip the pieces over when they have started to brown, then add 1/4 cup of chicken or vegetable broth. Cook until the chicken broth has been absorbed and evaporated. By this time, the eggplant slices should be thoroughly cooked.
Spread a film of olive oil onto a baking tray. Place the slabs of eggplant on the tray, then stack them each with a single slice of cooked puffball, and a slice or two of cooked shallot. Sprinkle on top a bit of mozarella and parmesan cheese, oregano and dill. Bake at 350ºF until the cheese melts (about 10 minutes.) Place on a presentation dish and serve. Should make about a dozen, enough to feed four as a main course, or 8-12 as a side dish.