Preheat oven to 425°F. Lay eggplant slices on paper towels and salt both sides; let stand for 30 minutes. Rinse and pat dry. Brush both sides with olive oil (1/4 cup total) and place on 2 baking sheets. Bake 20 minutes, turning slices and rotating pans halfway through. Remove eggplant from oven and let cool. Reduce oven temperature to 350°F.
In a large skillet with a lid, heat remaining oil. Add onion, mushrooms, zucchini and squash, season with salt and pepper and cook, covered, over low heat until soft, about 10 minutes. Remove lid, turn heat to high and cook until liquid evaporates. Add garlic, oregano, tomatoes and puree, and parsley and cook until sauce has thickened, about 10 minutes. Check seasoning and add more salt and pepper if necessary.
Mist a 13-by-9-inch baking dish with cooking spray. Cover bottom of dish with eggplant rounds, spoon half of vegetable sauce on top and sprinkle with 2 Tbsp. Parmesan. Repeat with remaining eggplant and sauce and 2 Tbsp. Parmesan, finishing with a top layer of eggplant.
Melt butter in pan and whisk in flour. Cook for 2 minutes. Whisk in milk and cook, stirring, until mixture comes to a boil. Reduce heat and simmer until sauce thickens, about 5 minutes. Add 1/2 cup Parmesan and yogurt and stir until smooth. Whisk in eggs and yolks.
Pour cheese sauce over eggplant. Place on a foil-lined baking sheet and bake until heated through and top is browned, about 70 minutes. Let rest for 20 minutes before slicing and serving.