Put the buttermilk and breadcrumbs in two small bowls and set aside.
Dip the eggplant slices in buttermilk, letting the excess drip back into the bowl. Then dredge them in the breadcrumbs. Repeat the process two more times, a total of 3 times, then set aside on a rimmed baking sheet or large plate.
In a medium skillet, heat the vegetable oil to 350°F or until the oil sizzles if you throw breadcrumbs in. Once heated, fry the eggplant for 2 minutes on each side, or until golden brown.
In a medium saucepan, add the wine, capers, garlic and heavy cream and season with salt and pepper. Bring to a simmer over moderate heat and cook until the liquid is reduced by half. Add the tomato and crabmeat, then reduce the heat to low and simmer for 10 minutes.
Arrange a slice of eggplant on each plate. Pour over some of the sauce over the top and then top with a second piece of eggplant and more sauce. Garnish with parsley and serve right away.