Bring a large (4-6 quarts) pot of water to a boil and add 3 tablespoons salt.

Slice eggplant in half lengthwise, then slice each in half again. Slice crosswise into ½-inch thick pieces.

Whisk together egg, 1 teaspoon salt, pepper, garlic and oregano in a large bowl. Add eggplant and toss to coat. Dredge eggplant in the panko crumbs on a cookie sheet.

Place a large rimmed baking sheet on the oven rack and heat for 10 minutes.

Pour enough olive oil over the bottom of the preheated baking sheet to cover to a depth of 1/4-inch. Lay eggplant on the pan and roast 12 minutes. Flip eggplant slices over with a spatula, and roast an additional 10 minutes or until eggplant is tender and crust is golden. 

Meanwhile, heat a medium saucepan over moderate heat. Add butter and heat until foaming subsides and butter is melted. Whisk in flour; cook 1 minute. Slowly pour in milk while whisking. Bring to a simmer, then lower heat and cook 5 – 10 minutes, whisking occasionally, until thickened. Remove from heat and stir in 1/4 cup of the Parmesan, 1/2 teaspoon salt and freshly ground black pepper.

Cook pasta according to package directions, until al dente. Drain.