1 tbsp. olive oil
3 medium eggplants (about 3 lbs.),
cut lengthwise into 1⁄4"-thick slices
Kosher salt and freshly ground black pepper, to taste
1 cup (2 sticks) plus 3 tbsp. butter
2 lbs. button mushrooms, thickly sliced
2 cloves garlic, finely chopped
1⁄4 cup dry red wine
1⁄4 cup chopped flat-leaf parsley leaves
2 tbsp. finely chopped basil leaves
1 tbsp. finely chopped oregano leaves
1 tsp. ground cinnamon
1 6-oz. can tomato paste
1 1⁄4 cups grated parmigiano-reggiano
3⁄4 cup bread crumbs
4 beaten eggs plus 4 egg yolks
1⁄2 cup flour
2 1⁄2 cups milk