In a large bowl, whisk together the flour, sugar, salt, and dry yeast.
Add the warm milk, eggs, and vanilla extract, and begin to combine the dough.
Melt the butter and gradually add it to the dough, 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.
Meanwhile, peel and dice the peaches, then mix them with the peach jam.
On a lightly floured surface, roll out the dough to about 1 cm thick. Use a large glass or cookie cutter to cut out circles.
Place a small spoonful of the peach 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, heat the oil to 350°F (180°C). Test the oil temperature with a small piece of dough – if it immediately floats to the surface 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 excess oil, then let them cool slightly.
Generously dust with powdered sugar and serve warm.
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