Preheat the oven to 390°F/ 200°C and line a baking pan with baking paper.
In a mixing bowl, combine the flour with the xanthan gum, salt, Erythritol, and baking powder.
Melt the butter over low heat, add the vanilla, and fold in the flour mixture, mixing constantly to incorporate. Keep mixing with a spatula or wooden spoon for 2-3 minutes until a dough is formed, and you can easily shape it into a ball that holds its shape. Add a little water if needed, one tablespoon at a time.
Transfer into a bowl and allow to reach room temperature before adding the egg and mixing it with an electric mixer until an elastic dough is formed.
Cover with a tea towel and let it rest for 15-20 minutes.
Shape into bite-sized balls with your hands and place on the prepared baking pan.
Bake for 10 minutes at 390°F/200°C then lower the temperature to 355°F/180°C and continue to bake for 5-10 minutes until they have risen into golden donut balls.