Activate the Yeast: Warm the milk until it is comfortably warm but not hot. Combine it with the yeast and a small portion of the sugar. Let it sit for about 5–10 minutes, until foamy.
Make the Dough: Add the eggs, melted butter, vanilla extract, remaining sugar, and salt to the yeast mixture. Gradually add the flour and mix until a soft dough forms.
Knead the Dough: Knead the dough on a lightly floured surface for about 8–10 minutes, or until smooth and elastic. Avoid adding too much flour.
First Rise: Place the dough in a lightly greased bowl and cover it. Let it rise in a warm, draft-free area for about 45–60 minutes, or until doubled in size.
Roll and Cut: Gently punch down the dough and roll it out to about ½-inch thickness. Use a round cutter to cut the donuts and transfer them to a parchment-lined baking sheet.
Second Rise: Cover the shaped donuts loosely and let them rest for about 20–30 minutes, until they look light and puffy.
Heat the Oil: Heat the vegetable oil in a deep pot to approximately 350°F (175°C). Maintain a steady temperature while frying.
Fry the Donuts: Carefully place the donuts into the hot oil in small batches. Fry for about 1–2 minutes per side, or until golden brown.
Drain and Coat: Remove the donuts with a slotted spoon and place them on a paper towel-lined plate. While still warm, gently roll them in granulated sugar.
Prepare the Cream: Beat the chilled heavy cream, powdered sugar, and vanilla extract until smooth and stiff peaks form. Transfer the cream to a piping bag fitted with a small round tip.
Fill the Donuts: Once the donuts have cooled, make a small opening on the side of each donut. Insert the piping tip and gently fill each donut with the vanilla cream.
Serve: Arrange the cream-filled sugar donuts on a serving plate and serve fresh.