Step 1
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.
Step 2
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.
Step 3
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.
Step 4
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.
Step 5
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.
Step 6
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.
Step 7
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.