Whisk together the egg, milk, and vanilla and vanilla paste.
In a separate bowl, whisk together the flour and yeast. Slowly stir the milk mixture into the dry ingredients until a dough forms.
Flour a counter top and knead, adding more flour a tablespoon at a time as needed, until a very slightly sticky, elastic ball of dough forms. Place in a buttered bowl, cover with a towel and allow to rise, about 1 hour or until it has doubled in bulk.
Punch down, and roll it out on a floured surface until it is about 1/2 to 2/3 of an inch thick. If you have a doughnut cutter, use that to cut out the doughnuts. If not, use a small cutter to make doughnut "holes" or improvise using a drinking glass or large cookie cutters and smaller cutters to make the holes. You could also make round, biscuit sized doughnuts with no holes.
Place the doughnuts on 2 greased cookie sheets, cover with a towel and allow to rise an additional hour. Preheat the oven to 375. Melt the additional 3 tablespoons of butter. Pour some (granulated) sugar on a plate. Bake the doughnuts exactly 6 minutes, then remove to a wire rack.
Brush each doughnut with melted butter and dip in sugar. Serve hot, although they are still pretty good at room temperature.
Store up to 1 day (but eating them immediately is preferred) in a air tight container.