In a large mixing bowl, whisk together the flour, granulated sugar, yeast, and salt.

Make a well in the center and add the warm milk, softened butter, egg, and vanilla extract.

Stir the mixture until a shaggy dough forms. Use your hands or a stand mixer to knead the dough for about 5 to 7 minutes until it is smooth and stretchy.

Place the dough in a lightly oiled bowl, cover it with a clean kitchen towel, and let it rise in a warm place for about 1 to 1.5 hours, or until it doubles in size.

Punch down the dough to release air. Turn it out onto a floured surface and roll it to about 1/2-inch thickness.

Use a round cookie cutter or a glass to cut out circles about 2.5 inches wide. Place the circles on a parchment-lined baking sheet.

Cover the circles and let them rise again for another 30 minutes until they look puffy.

Heat the vegetable oil in a deep pot until it reaches 350°F (175°C). Carefully drop 3 or 4 donuts into the oil at a time.

Fry for about 2 minutes per side until they are deep golden brown. Remove them with a slotted spoon and drain on paper towels.

Once cool enough to handle, use a small knife or a skewer to poke a hole into the side of each donut.

Fill a piping bag with jelly and squeeze about a tablespoon of jam into the center of each donut.

Toss the filled donuts in powdered sugar or sift the sugar over the top before serving.