Heat 2 tablespoons olive oil in a skillet over medium-high heat. Cook and stir diced onion with 1 teaspoon salt until onion is slightly translucent, about 5 minutes. Add eggplant, reduce heat to medium, and cook and stir until eggplant is opaque and softened, 3 to 5 minutes more.

Stir garlic, 1 teaspoon black pepper, cumin, cinnamon, and oregano into eggplant mixture; cook and stir until fragrant, about 1 minute. Add tomato paste; cook and stir until heated through, about 2 minutes. Transfer mixture to a plate and cool completely. Cover plate with plastic wrap and refrigerate until chilled, at least 15 minutes.

Whisk 1 1/2 tablespoons olive oil and flour together in a saucepan over medium-high heat until golden and bubbling, 2 to 3 minutes. Pour milk into butter-flour mixture, whisking constantly as it simmers and forms a smooth, thick sauce, 3 to 5 minutes. Season with nutmeg, a pinch black pepper, and a pinch cayenne pepper. Remove from heat and stir Parmesan cheese into milk mixture until melted.

Combine lamb, chilled eggplant mixture, and a pinch of salt together in a bowl. Form into four patties and place on a plate. Cover the plate with plastic wrap and refrigerate until just before cooking.

Place tomato slices in a shallow bowl. Season tomatoes with chopped mint, a pinch of salt, and a pinch of black pepper. Pour rice vinegar over the top and toss to coat.

Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Cook burgers until slightly pink in the center, about 4 minutes per side. An instant-read thermometer inserted into the center should read 140 degrees F (60 degrees C) for medium.

Spread desired amount of cheese sauce onto both sides of each hamburger bun. Place burger on bottom bun, place 2 tomato slices atop the burger, and place top half of the bun over the tomatoes.