Clean, peel and wash the carrots, cut them in half lengthwise and slice them. Clean and wash leek, cut into halves lengthwise and cut into rings. Cook rice in plenty of boiling salted water for about 20 minutes.
Add carrots after 10 minutes, leek after 15 minutes. Drain and allow to drain. Clean, dab dry and halve the mushrooms. Wash the meat, dab dry and cut into cubes. Heat oil in a coated pan and fry meat until golden brown.
Season with salt and pepper and remove. Fry the mushrooms in the frying fat and take them out. Mix rice and vegetable mixture with meat and mushrooms and put into 4 greased casseroles (250 ml each).
Whisk milk and eggs and season with salt, pepper and nutmeg. Distribute into the moulds and sprinkle with cheese. Bake in the preheated oven (electric cooker: 200 °C/ convection oven: 175 °C/ gas: level 3) for 20-25 minutes.
In the meantime, peel and finely dice the onion and garlic. Put them in a pot with the tomatoes, heat them up and let them simmer for 10 minutes at low heat. Season to taste with salt, pepper and sugar.