In a large bowl, combine the flour, sugar, salt, and dry yeast.
Add the warm milk, eggs, and vanilla extract, and begin to mix the dough.
Melt the butter and gradually add it to the dough while kneading continuously until you have a smooth, elastic dough.
Cover the dough with a kitchen towel and let it rise in a warm place for about 1 hour, or until doubled in size.
Mix the cream cheese with the grated cheese to create a creamy, savory filling.
On a lightly floured surface, roll out the dough to about 1.5 cm (approximately 1/2 inch) thick. Use a large glass or cookie cutter to cut out circles.
Place a small spoonful of the cheese filling in the center of each dough circle, then top with another dough circle and press the edges together to seal.
In a deep pot or fryer, heat the oil to 350°F (180°C). Check the oil temperature with a small piece of dough – if it immediately floats to the top and gently fries, the oil is ready.
Fry the filled doughnuts for 2-3 minutes on each side, until golden brown.
Remove the fried doughnuts and place them on a paper towel-lined plate to absorb any excess oil. Let them cool slightly.
Sprinkle with powdered sugar, or if you prefer a more savory flavor, grate a little extra cheese over them before serving.
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