How to Make It
Heat oil in a large skillet over medium-high. Cook eggplant, onions, and carrots, stirring, until tender and starting to brown, 10 to 15 minutes. Stir in broth, tomato paste, parsley, pepper, and 3/4 teaspoon salt; bring to a simmer. Cook, stirring often, until thickened, about 15 minutes. Cover and keep warm.
Preheat broiler with rack 5 inches from heat. Place potatoes in a medium pot and cover with salted water by 1 inch. Bring to a boil and cook until very tender, about 15 minutes. Drain and return potatoes to pot; mash with cheese, milk, and remaining 1/4 teaspoon salt.
Spread potatoes evenly over eggplant mixture. Broil until golden brown, about 5 minutes. Top with parsley.