Clean, wash and slice the aubergines. Heat 5 tablespoons of oil bit by bit in a frying pan. Fry the aubergines in it in portions on each side for 2-3 minutes until golden brown. Remove and drain on kitchen paper.
Wash and slice the tomatoes. Cut mozzarella into slices. Wash the thyme, shake dry and remove the leaves from 3 stems. Peel garlic and cut into thin slices.
Preheat oven (electric cooker: 200 °C/circulating air: 180 °C/gas: see manufacturer). Spread a casserole dish with a little oil. Place aubergines, tomatoes and mozzarella alternately in the dish. Always season with thyme, garlic, salt and pepper. Finish with mozzarella. Bake in a hot oven for about 30 minutes.
Grate the Parmesan finely and sprinkle it on the casserole dish about 10 minutes before the end of the cooking time. Remove the aubergine casserole. Garnish with the rest of the thyme and serve. Rice or polenta are suitable as gluten-free side dishes. But be careful! This means it is no longer a low-carb recipe.