Place eggplant in non-reactive bowl. Toss with 1 teaspoon salt and set aside for 20 minutes. Remove to a clean kitchen towel and tightly twist to squeeze out all moisture.
Heat 2 tablespoons of oil in sauté pan. Add eggplant and shallots and sauté over medium heat until very tender. Remove from heat. Peel potatoes.
Using a mandoline, shred potatoes into a clean kitchen towel. Tightly squeeze out moisture. Heat remaining 1/4 cup of oil in a 9-inch ovenproof, nonstick sauté pan over medium heat. Using a spatula, evenly press half of the potatoes into the pan. Season with salt and pepper.
Spread the reserved eggplant over the top. Pat remaining potatoes evenly over the eggplant. Again season to taste. Cook, until bottom is golden. Carefully turn and dab the crust with bits of butter. Cook until bottom begins to crisp.
Place in a 350 degree oven and bake for about 30 minutes or until potatoes are cooked and tart is golden and crisp.
Remove from oven and allow to rest 5 minutes. Cut into 6 wedges.