In a bowl, mix the yeast, a pinch of the sugar and 1/2 cup of the water and let stand until foamy. Stir in the egg, salt and the remaining sugar and 1 1/2 cups of water. Add 5 cups of the flour and stir to form a soft, sticky dough. Stir in the stick of softened butter. Add the remaining 1 cup of flour and stir. When the dough becomes too stiff, scrape it out onto a work surface and knead until smooth; add a few tablespoons of flour if necessary to keep the dough from sticking. Transfer the dough to a lightly buttered bowl, cover with plastic wrap and let rise in a draft-free place until doubled in bulk, 1 hour.
Punch down the dough and knead 2 or 3 times; cover and let rest for 15 minutes. On a very lightly floured surface, roll out the dough 3/8 inch thick. Using a 2 3/4-inch round cutter dipped in flour, stamp out 24 rounds as close together as possible; if necessary, reroll and cut out some of the scraps. Reserve the scraps.
Melt the remaining 10 tablespoons of butter in a small saucepan; set aside 2 tablespoons. Dip both sides of each round in the remaining butter, letting any excess drip into the pan. Fold the rounds in half and set, folded side down, in a 9-by-13-inch glass baking dish. Toss the reserved scraps with any leftover butter and arrange them in a single layer in an 8-inch square baking dish. Cover both dishes with plastic wrap and let the rolls rise in a draft-free spot until nearly doubled in bulk, about 1 hour.
Preheat the oven to 425°. Remove the plastic and bake the rolls for 30 minutes, or until golden, shifting the pans halfway through baking.
Remove the rolls from the oven and brush the tops with the reserved 2 tablespoons of butter. Let cool for 10 minutes. Invert the rolls onto a wire rack, set a platter on top and invert again. Serve warm or at room temperature.
The baked rolls can be stored overnight in an airtight container or frozen for 1 month and rewarmed.