Preheat the oven to 200C/180C/Gas 6. Grease a deep baking dish with the butter.
Alternately layer the parsnips, potatoes, onion and garlic in the dish, sprinkling between each layer with a mixture of all three cheeses and a few thyme leaves. Continue to layer until all of the ingredients have been used up, reserving a few thyme leaves and some cheese.
Place the cream in a saucepan with the reserved thyme leaves, nutmeg, salt and white pepper and heat for a few minutes, until warm. Pour the cream into the baking dish, making sure the cream has properly soaked through all the layers.
Scatter the reserved cheese over the top, cover with kitchen foil and bake for 1½ hours, or until the vegetables are tender. Remove the foil and crisp the top for a further 20 minutes. Leave to stand for 30 minutes before serving. Keep the oven on.
For the venison, blitz the oil, oregano and garlic purée in a small food processor. Season with salt and pepper. Rub the paste all over the venison rump.
Place a frying pan over a high heat, add the rump and seal on all sides. Place the rump in a roasting tin and cook in the oven for 8–10 minutes. Leave to rest for a further 10 minutes before carving.
Transfer the juices from the roasting tin to a saucepan and place over a medium heat. Add the red wine and simmer until reduced by half. Whisk in the butter until you have a thick consistency. Season with salt and pepper.
Cut the venison into slices and place on warmed plates with the gratin on the side. Spoon the sauce over the venison and serve immediately.