Pour 4 tablespoons of warm water into a small mixing bowl. The water should be hot, but not so hot that you can bear to put your fingers in it. Add ½ of the sugar and yeast, and stir to dissolve.
Wait about ten minutes for the yeast to begin to grow.
In a large mixing bowl add the flour, salt, remaining sugar and the water and yeast mixture and begin mixing. Use your hands for this, as you really get the perfect feel for the consistency of the dough. Add the butter, vanilla and eggs. Mix well
Knead the dough on a clean, dry, flat surface. Sprinkle your work surface with a handful of flour, put your dough on top, and start kneading. Add bits of flour if necessary to keep the dough from sticking to your hands, or surface.
Keep kneading until the dough is ready. This may take 8 to 10 minutes. It will be quite elastic. It should not be too dry, however, it should still give and stretch easily without tearing.
Transfer to a lightly oiled bowl, turn once to coat, and cover with plastic wrap. Let rise in a warm place until doubled in bulk, about 2 hours.
Turn the dough out onto a lightly floured work surface and knead for 30 seconds.Cover with a towel and let rest for 10 minutes. Punch the dough down and divide into 9 equal size pieces.
Shape each piece into a smooth ball and place into a square 8 x 8 or 9 or 10 inch round baking dish lightly greased, spacing evenly. Cover and let rise in a warm place for 20-30 minutes.
Preheat the oven to 400 F. Brush the tops of the rolls with the beaten egg and melted butter. Bake for 15-20 minutes, or until the tops are golden brown and the rolls are baked through. Sprinkle with sugar on top and let them cool slightly before serving.