In a large bowl, whisk together the flour, sugar, salt, and active dry yeast.
Pour the warm milk into the dry ingredients, then add the egg, melted butter, and vanilla extract.
Mix the ingredients until combined, then knead the dough for 8-10 minutes until it is smooth and elastic.
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 1/2 inch (1 cm) thickness. Use a glass or doughnut cutter to cut out circles.
Place the cut doughnuts on a baking sheet lined with parchment paper, and cover with a clean kitchen towel. Let them rise for another 30 minutes.
In a large pot, heat the oil to about 340°F (170°C). Test the oil by dropping in one doughnut: if it immediately rises to the surface and bubbles gently, the oil is ready.
Fry the doughnuts for 1-2 minutes per side, or until golden brown. Use a slotted spoon to remove them, then place them on a paper towel-lined plate to drain excess oil.
Make the fry sauce dip: combine the ketchup, mayonnaise, vinegar, sugar, and Hungarian Sweet Paprika in a small bowl.
Serve the doughnuts dusted with powdered sugar, and dip them into the fry sauce for a sweet and savory contrast.
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