Finely chop the onion and garlic. In a heatproof container, combine the chopped onion and garlic with ground beef, canned tomatoes, and 2 tablespoons of miso. Mix gently, then microwave at 600W for 8 minutes. Once heated, stir the mixture to combine.
Dice the eggplant into bite-sized pieces. In a frying pan, heat olive oil over medium heat. Add the eggplant and cook until it is evenly browned on all sides, then remove it from the pan.
In the same frying pan, combine rice, milk, the remaining 1 tablespoon of miso, pepper, and 1 tablespoon of olive oil. Cook over medium heat until everything is well blended.
Once the mixture is evenly combined, add the eggplant followed by the meat mixture and top with shredded cheese. Cover the pan with a lid and cook over low heat until the rice is tender and the cheese melts.
Bring the frying pan to the table and serve the dish directly from it.