Preheat oven to 450. Cover bottom of two sheet pans with aluminum foil. Coat foil and both sides of each piece of eggplant thoroughly in olive oil. Place eggplant slices into a singe layer across the two sheets.

Set up oven racks to lower middle and upper middle positions. Put eggplant in oven and roast for 20 minutes, switching the rack position and rotating each pan after 10 minutes for even cooking. Remove and allow to cool at least 10 minutes. Reduce oven to 375.

While you are roasting the eggplant, start the sauce. Heat olive oil over medium-high heat until shimmering. Add onion and cook until translucent. Add garlic and cook 30 seconds until fragrant. Add tomato paste, anchovy paste, and vegetable base. Mix in and cook about 30 seconds. Add dry sherry. Cook about 2 minutes. Add tomatoes and all other seasonings. Bring to a simmer and then reduce to medium-low heat. Cover. Cook, stirring occassionally, about 20 minutes.

Grease an oval or rectangular baking dish well with cooking spray. Place 1/2 cup of sauce into the bottom and spread across the whole bottom. Place a single lawyer of eggplant over the sauce. Add another 1/2 cup of sauce and 1/3 of the cheeses.

Place another layer of eggplant, 1/2 cup of sauce, and 1/3 of the cheese. Continue with the final layer of eggplant and top with remaining sauce and cheese. Place into the oven, uncovered, and bake for 25 - 30 minutes. Remove and let sit for 10 minutes before serving.