Step 1

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.

Step 2

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.

Step 3

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.

Step 4

Preheat the oven to 425°. Remove the plastic and bake the rolls for 30 minutes, or until golden, shifting the pans halfway through baking.

Step 5

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.