In a small bowl, combine warm milk, yeast, 2 tablespoons sugar, and 2 tablespoons flour. Whisk well and set aside to proof for 5 minutes, or until foamy.
In another bowl, combine butter, egg yolks, vanilla, and 2 tablespoons sugar. Mix will the yeast mixture. Then gradually add the rest of the flour into bowl.
Knead the dough until smooth. If the mixture is too sticky, sprinkle a little bit of flour at a time. Return the dough to the bowl, cover, and set aside to rise in a warm spot for about 1 1/2 hours or until doubled in size.
Place the dough onto a floured surface. Gently roll the dough into a log and cut into 6- 8 equal portions.
Form each portion into a ball and place in a greased muffin tin. Cover and let it rise again for about 45 minutes.
Brush the tops with melted butter and bake at 400°F for 15-20 minutes, until golden brown. Set aside to cool.
Using a piping bag, fill each donut muffin with vanilla pudding.
Place chocolate chips and butter in a bowl and microwave at 30 second intervals until fully melted. Mix well. Dip the tops of each donut muffin and set aside to cool.