In a large mixing bowl, whisk together the gluten-free flour, baking powder, salt, and granulated sugar until well combined. This ensures even distribution of dry ingredients.

In a separate bowl, beat the egg, then add the milk, melted butter, and vanilla extract. Mix until fully combined.

Pour the wet ingredients into the dry ingredients and stir until a soft dough forms. Be careful not to overmix, as this can make the donuts tough.

Cover the bowl with plastic wrap and let the dough rest for about 15 minutes. This allows the flour to hydrate and the dough to become more pliable.

On a lightly floured surface (using gluten-free flour), roll out the dough to about ½ inch thick.

Using a donut cutter, cut out the donuts. If you don't have a donut cutter, you can use two round cookie cutters of different sizes.

Place the cut donuts on a parchment-lined baking sheet and let them rest for another 10 minutes. This helps them rise slightly before frying.

In a deep fryer or large pot, heat the vegetable oil to 350°F (175°C). Use a thermometer to ensure the oil is at the right temperature for frying.

Carefully drop a few donuts into the hot oil, being careful not to overcrowd the pot. Fry for about 2-3 minutes on each side, or until golden brown.

Use a slotted spoon to remove the donuts from the oil and place them on a cooling rack to drain excess oil.

Once the donuts are cool enough to handle, use a piping bag fitted with a small round tip to fill each donut with fruit preserves. Insert the tip into the side of the donut and squeeze gently.

After filling, dust the donuts with powdered sugar for a sweet finish.