In a medium saucepan, heat the olive oil over moderately low heat. Add the onion and cook, stirring occasionally, until translucent, about 5 minutes. Add the tomatoes, tarragon, 3/4 teaspoon of the salt, and the sugar and simmer, stirring occasionally, for 25 minutes. Stir in 1/4 teaspoon of the pepper.
Meanwhile, heat the oven to 350°. In a medium bowl, combine the bread crumbs, Parmesan, and the remaining 1/2 teaspoon salt and 1/4 teaspoon pepper.
Dip each slice of eggplant in the eggs and then in the bread-crumb mixture, coating well. In a large frying pan, heat about half an inch of cooking oil over moderate heat until very hot. Fry the eggplant in the hot oil, in batches, turning once, until golden and cooked through, 1 to 2 minutes per side. Drain on paper towels.
Arrange half of the eggplant slices in a single layer on a baking sheet. Put a slice of goat cheese on top of each and then top with the remaining eggplant slices. Bake until the cheese melts, about 10 minutes. Put the sauce on plates and top with the eggplant.