In a large mixing bowl, whisk together 3 cups of the all-purpose flour, sugar, salt, and active dry yeast.
In another bowl, stir together the milk, melted butter, and vanilla extract. Add to the dry ingredients.
Add the egg and using a wooden spoon, stir everything together until a soft, sticky dough is formed and gathered in the center of the bowl.
Cover the bowl and let the dough rest for about 5 minutes. Note that the dough will be very sticky but It will gradually lose its stickiness once it’s kneaded.
Generously dust a flat working surface with some of the remaining 1/2 cup of flour. Turn the dough over to the floured surface.
Knead the dough, adding flour gradually as needed if the dough sticks to the working surface or hands. You do not have to use the entire 1/2 cup of flour. Add only as needed to make the dough manageable and to keep from sticking. If the dough feels dry and stiff, you have added too much flour.
Continue to knead until the dough is smooth and elastic. To test, take a small portion of dough, stretch it out to form a little square using your fingers. The dough should form a translucent window in the center before it tears.
Shape the dough into a ball, brush lightly with oil and put in a bowl. Cover the bowl and let the dough rise for about 1 1/2 to 2 hours or until double in size.
Using a rolling pin, roll the dough to 1/4 inch thickness. Using a 2 1/2-inch diameter donut cutter, cut 14 circles from the dough.
Arrange them in a single layer on a parchment-lined baking sheet lined with parchment paper along with the donut holes. Cover loosely with plastic wrap or a clean kitchen towel and let rise for another hour.
In a deep fryer or a medium heavy-bottomed saucepan, heat about 5 cups of oil to 350 F.
Carefully lower the donuts, 2 or 3 at a time, and cook until the underside is lightly golden. Turn over donuts and cook on the other side. Cook the donut holes for about 1 to 2 minutes or until lightly golden.
Remove the donuts from the pan and drain on a baking sheet lined with paper towels.
Dip one part of the donuts in the vanilla glaze to coat and top with chopped Oreo cookies.