Arrange three small bowls on a work space and slice the eggplant into ½-inch rounds. Whisk the eggs together with the pesto in the first bowl. Stir together the flours, salt and pepper in the second bowl. Combine half of the cheeses, breadcrumbs and oregano in the third.
Heat a griddle over medium heat, and lightly oil the pan.
Dip an eggplant slice into the flour, followed by the egg-pesto mixture, and finally the cheeses. Place on the griddle. Repeat with enough eggplant to fill the griddle (I can fit around 8 slices of various sizes on mine). Cook for 5 minutes per side, until the eggplant surfaces are lightly browned. Stack and repeat with the remaining eggplant. For a 8 ½" X 12" baking pan, I typically use 16 slices.
Make Ahead Tip: At this point, the tomato sauce can be poured into a jar and the eggplant parmesan carefully stacked in ziplock baggies. Tuck into the refrigerator for up to 24 hours.
Preheat the oven to 400˚F. Arrange the eggplant, tomato sauce and remaining cheese in layers as in a lasagna, spreading a thin layer of tomato sauce on the bottom of a 8 ½" X 12" or 9" X 13" baking pan. Next arrange a layer of eggplant parmesan slices and sprinkle with ⅓ of the remaining cheese. Repeat the order, starting with half of the remaining tomato sauce, a final layer of eggplant slices, half of the remaining cheese, and the remaining tomato sauce.
Bake for 30 minutes. Add the remaining cheese on top and bake another 15 minutes.
Cool slightly before serving.