In a small bowl, mix the yeast, 1 tsp sugar and 1/4 cup lukewarm water. Set it aside until foamy.
In a large mixing bowl, combine the flour, salt and 1/4 cup sugar. Pour in the yeast mixture, melted butter, egg and the milk, use a mixing spoon and keep stirring until everything well-combines, for about 5 minutes. (The dough will be super sticky, pretty similar to cake batter).
Cover the bowl and let the dough proof until double its size. Prepare and line a baking pan with parchment paper.
Since the dough is super-sticky, make sure the work table is well-floured, pour the dough on the table; it would spread out by itself since it's very soft and lumpy. Do not knead or pat it too hard (as it will deflate the air bubbles and your donuts won't be that airy).
Pat the dough lightly to about 2 cm. Dip the round cookie cutter with flour first and cut the dough, use a spatula to help you transferring the round dough onto a large baking tray. Cover the dough and let it rest for 20 minutes.
Pre-heat the oven to 375 degree F. Brush the donuts with some melted butter and baked the donuts until lightly brown. It will take about 12 minutes (depend on your oven). Cool the donuts on a rack.
Prepare a pipping bag fitted with a long round cake decorating tip. Fill it up with your favorite jam or custard and pipe it into the donuts. Lightly dust it with some powder sugar before serving.