Preparation

Make the bottom layer

• Position racks in the upper and lower parts of the oven, and heat the oven to 425°F. Line two large rimmed baking sheets with paper towels. Sprinkle both sides of the eggplant lightly with salt, transfer to the lined baking sheets, and let sit for 30 minutes. Line two rimmed baking sheets with parchment. Pat the eggplant slices dry with paper towels, brush both sides generously with oil, and place on the prepared baking sheets in a single layer. Bake the eggplant for 15 minutes on the lower rack. Flip the eggplant, brush generously with more oil, and continue roasting until the eggplant is soft and lightly brown in spots, 15 to 20 minutes more. Remove from the oven, and set aside to cool.
• Reduce the oven temperature to 400°F. Line a large rimmed baking sheet with parchment. Toss the tomatoes in a large bowl with the vinegar, oregano, 1 tsp. salt, and 1/2 tsp. pepper. Put the tomatoes skin side down and close to each other on the baking sheet, transfer to the upper rack of the oven, and roast until soft and dark brown in places, 38 to 40 minutes. When slightly cool, transfer to a blender and purée until smooth, scraping down the bowl as necessary, 1 to 2 minutes. Season to taste with salt and pepper, and set aside.

Make the filling

• Heat 1 Tbs. of the oil in a large skillet over medium heat until shimmering. Add the zucchini, 1/2 tsp. salt, and 1/2 tsp. pepper, and cook, stirring occasionally, until the zucchini softens and browns in spots, 5 to 6 minutes. Reduce the heat to low, add the tomato paste, and cook, stirring constantly, about 1 minute. Add the wine, bay leaves, and cinnamon, and cook, stirring frequently, until the wine mostly evaporates and the mixture thickens, 1-1/2 to 2 minutes. Transfer to a large bowl; discard the cinnamon stick and bay leaves.
• Put the onions and mushrooms in two separate bowls, and toss each with 1-1/2 Tbs. oil, 1 tsp. salt, and 1/2 tsp. pepper. Spread on two separate large rimmed baking sheets in a single layer. Transfer to the oven and roast, stirring occasionally, until the onion is soft and lightly brown in spots, and the mushrooms are mostly dry, about 1 hour (caution: a lot of steam will escape from the oven when you open the door). Remove from the oven to cool completely. When cool, transfer to the bowl with the zucchini mixture, and toss to combine. Set aside. Reduce the oven temperature to 375°F.

Make the topping and assemble the moussaka

• Bring a large pot of well-salted water to a boil. Add the potatoes and cook until fork-tender, about 15 minutes. Remove from the pot with a slotted spoon, and set aside. Add the carrots to the same pot, and cook until very tender, about 15 minutes. Drain the carrots. Put the potatoes and carrots through a ricer over a large bowl, or alternatively, mash with a potato masher until smooth. Add the garlic, broth, chives, and 2 Tbs. of the oil. Stir with a silicone spatula to combine.
• Position a rack in the center of the oven. Brush a 9×13-inch (3-quart) glass or ceramic dish with 1 Tbs. of the oil. Spread enough of the tomato sauce to coat the bottom of the pan (you may have some left over). Cover the sauce with the eggplant slices, overlapping slightly. Make a second layer with any remaining eggplant. Press down lightly. Spread the filling evenly over the eggplant, pressing down lightly.
• Top with the mashed potato-carrot mixture, spreading evenly over the filling. Brush with the remaining 1 Tbs. oil.  Put the dish on a rimmed baking sheet, and bake until the top is lightly golden and the filling is bubbling, 40 to 45 minutes. Remove the moussaka from the oven, and let rest for at least 20 minutes. Serve sprinkled with more chopped chives.