In a large mixing bowl, combine the warm milk, yeast, and 1 tablespoon of the sugar. Leave it covered for about 5 to 10 minutes until a frothy layer forms, indicating the yeast is activated.

Next, incorporate the remaining sugar, eggs, melted butter, vanilla extract, nutmeg, salt, and 2 cups of flour. Mix at low speed for around a minute, scraping down the bowl sides. Gradually add the remaining flour and increase the speed to medium until the dough starts to detach from the bowl sides, which should take about 2 minutes. If the dough feels excessively sticky, add flour a tablespoon at a time until it reaches a tacky consistency.

Knead the dough for 5 to 7 minutes, either using your mixer or by hand on a floured surface. If it clings, dust your hands or surface lightly with flour. The dough should be soft and springy; a gentle poke should show it bouncing back.

Lightly grease a large bowl and place your dough inside, ensuring all sides are coated with oil. Cover the bowl and let the dough rise in a warm area for 1.5 to 2 hours, or until it has doubled in size.

Once the dough has risen, gently punch it down to release the air. Transfer it to a floured surface and roll to approximately 1/2 inch thickness. Use a 3 to 3.5-inch doughnut cutter for shaping. Reroll the scraps if needed and set the cut doughnuts on parchment-lined trays. Cover lightly and let them rest for an additional 30 minutes while you heat the oil.

Heat vegetable oil in a deep pot to 375°F (191°C). Fry 2-3 doughnuts at a time for about 1 minute on each side until golden brown. Use a slotted spoon to move them to a cooling rack placed over a baking sheet, taking care to maintain the oil temperature and minimize splattering.

For the glaze, whisk together powdered sugar, cream (or milk), and vanilla until fully combined and smooth. While the doughnuts are still warm, dip them into the glaze, ensuring both sides are well coated. Allow them to rest on the rack for about 20 minutes to let the glaze firm up.