Make the rice porridge: in a large saucepan, cook the rice in 2 dl of water until it is absorbed. Add the milk, bring to a simmer, then cook for 40 minutes over very low heat, stirring often so the rice does not stick. It should be thick and creamy. Add the salt and nutmeg and leave to cool slightly.

Make the rye dough: mix the cold water and salt, then add both flours. Knead just enough to get a smooth, firm dough. Shape it into a log and cut it into 16 pieces.

Preheat the oven to its highest setting (250 to 275 °C). On a floured work surface, roll each piece into a very thin oval about 15 cm long. It should be almost see-through.

Place 2 heaped tablespoons of rice porridge in the centre of each oval, leaving a 1.5 cm border. Fold the edges over the rice and crimp them between your thumb and index finger all the way round, leaving the rice clearly visible in the middle.

Place the pies on a baking tray lined with baking paper. Bake for 12 to 15 minutes, until the rice has golden spots and the edges are crisp.

As soon as they come out of the oven, heat the melted butter with the milk and brush the pies generously. Cover them with a tea towel for 10 minutes so the crust softens.

Make the munavoi: boil the eggs for 10 minutes, peel them and chop them finely. Mix them with the soft butter, and season with salt and white pepper. Serve the karjalanpiirakka warm, topped with a spoonful of munavoi.