In a large bowl, whisk together the all-purpose flour, granulated sugar, salt, and dry yeast.
Pour the warm milk into the dry ingredients, then add the egg, melted butter, and vanilla extract.
Mix all the ingredients together, then knead the dough for 8-10 minutes until it becomes elastic and smooth.
Cover the dough and let it rise in a warm place for about 1 hour, or until doubled in size.
On a lightly floured surface, roll out the dough to about 1 cm thickness. Sprinkle the chocolate chips over the dough and gently knead them in.
Use a glass or doughnut cutter to cut out circles from the dough and place them on a baking sheet lined with parchment paper.
Cover the doughnuts with a clean kitchen towel and let them rise for another 30 minutes.
In a large pot, heat the oil to about 340°F (170°C). Drop a doughnut into the oil as a test; if it immediately floats to the surface and bubbles gently, the oil is ready.
Fry the doughnuts for 1-2 minutes per side, until golden brown. Use a slotted spoon to remove them and place them on paper towels to absorb excess oil.
Once the doughnuts have cooled slightly, sprinkle them with powdered sugar or prepare a chocolate glaze.
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