1. Preheat oven to 400 degrees, with racks in middle and upper third. Divide eggplant between two rimmed baking sheets. Toss each with 1 tablespoon oil and season with salt and pepper. Bake until tender, 20 minutes, stirring eggplant and rotating sheets halfway through. Let sheets cool on wire racks.

2. Meanwhile, in a large skillet, heat 1 tablespoon oil over medium-high. Add beef and cook, stirring occasionally, until cooked through, 4 minutes. Add allspice, mint, and oregano and season with salt and pepper. Cook, stirring occasionally, until fragrant, 3 minutes. Transfer beef to a medium bowl.

3. Spread 1 cup sauce in an 8-inch square baking dish. Top with 2 noodles. Layer with half the beef, then 1 cup sauce and 1/4 cup Parmesan. Top with 2 more noodles, half the egplant, 1 cup sauce, and 1/4 cup Parmesan. Repeat layering with remaining ingredients, reserving the last 1/4 cup Parmesan. Cover lasagna loosely with foil and bake on top rack until sauce is bubbling, 35 to 40 minutes. (Place a rimmed baking sheet on middle rack to catch any drippings.) Remove lasagna from oven and heat broiler. Sprinkle with reserved Parmesan and broil until cheese is browned and bubbling, 3 to 5 minutes. Let cool 20 minutes before slicing.