Heat the milk in the microwave for about 20 seconds, just until it's slightly warmer than your body temperature. Add the yeast and a small pinch of sugar to the milk, then let it sit for five minutes to activate.
Melt the butter using either the microwave or a saucepan, then transfer it into the bowl of your stand mixer. Add the flour, sugar, nutmeg, vanilla extract, salt, and eggs. Mix on high speed until it comes together into a dough. Swap the beater for a dough hook and knead the mixture for 5 minutes. The dough will be slightly sticky, but that's perfectly fine.
Transfer the dough to a bowl greased lightly with oil, then cover it with a tea towel or plastic wrap. Set the bowl in a warm spot and let it rise for at least an hour, or refrigerate it overnight. The dough should double in size.
Turn the dough out onto a well-floured surface and knead it until it becomes less sticky, adding more flour if needed. Roll it out to about ¼ inch thick and use a 4-inch round cookie cutter to shape the doughnuts. Arrange them on a floured baking tray, leaving enough space between each one to prevent them from sticking together as they rise. Place the tray in a cold oven, then position a bowl of boiling water at the bottom of the oven to create steam and help the doughnuts rise. Close the oven door and let them rise for 30 minutes.
After the doughnuts have finished rising, take them out of the oven. Fill a large saucepan halfway with sunflower oil and heat it over medium-high heat to 190°C (365°F) using a candy or deep-frying thermometer. Gently lower a couple of doughnuts into the hot oil with a spatula. Fry each side for about 30 seconds, though the first one may take a little longer as you adjust the timing. Once golden, carefully remove the doughnuts with a spatula and place them on paper towels to drain excess oil. Continue frying the remaining doughnuts, adjusting the heat as needed to maintain the right temperature. Once all the doughnuts are fried, let them cool completely.
Use a knife to make a small hole on the side of each doughnut, then carefully pipe the cookies and cream filling inside. Take care not to overfill, as this could cause the doughnut to burst. Make sure to use a wide round nozzle for your piping bag, as the Oreo crumbs could clog a smaller one. Once all the doughnuts are filled, set them aside and prepare the glaze.