In a bowl, whisk together the flour, baking powder, and salt.
In another bowl, whisk the eggs with the sugar, then add the milk, melted butter, and vanilla extract.
Gradually add the dry ingredients to the wet mixture, and mix until you have a smooth, homogenous dough.
Cover the dough and let it rest for 30 minutes, allowing the baking powder to activate.
On a lightly floured surface, roll out the dough to about 1 cm thickness. Use a doughnut cutter or a glass to cut out circles.
In a deep skillet, heat the oil over medium-high heat.
Fry the dough circles until golden brown, about 2-3 minutes per side, watching carefully to prevent burning.
Remove the fried doughnuts and place them on paper towels to drain excess oil.
While still warm, dust the doughnuts with powdered sugar and serve fresh.
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