Wash the potatoes and boil them for about 25 minutes until soft. Cut the cauliflower into small florets and cook in boiling salted water for about 4 minutes, drain and rinse with cold water. Peel and slice the carrots.
Heat up the carrots, some salt and about 5 tablespoons of water, cover and stew for about 3 minutes. Heat the fat in a saucepan. Stir in flour and sauté briefly. Gradually stir the stock and milk into the roux.
Bring to the boil and simmer while stirring for about 5 minutes. Wash parsley, dab dry and chop the leaves of 2 stems finely. Peel onion and cut into small cubes. Heat oil in a pan and sauté onion for 2-3 minutes.
Add the mince and fry it. Stir in tomato paste and add 100 ml water. Add parsley and let the minced meat simmer for 3-4 minutes and season with salt, pepper and paprika. Drain the potatoes, rinse briefly with cold water and peel.
Spread about 3/4 of the potatoes in a greased high casserole dish (the bottom should be covered straight). If the potatoes are larger, you can also cut them in half. Spread half of the minced meat on the potatoes, then add about 2/3 of the béchamel.
Finish with carrots, cauliflower, the remaining potatoes and minced meat in rows and pour 1/3 of the béchamel over the cauliflower and between the rows. Grate the cheese and sprinkle over the cauliflower. Bake in a preheated oven (electric cooker: 200 °C/ convection oven: 175 °C/ gas: level 3) for about 20 minutes.
Garnish the finished casserole with parsley.