1. In a bowl, sprinkle yeast over 1 1/4 cups warm (about 110°) water; let stand until yeast is soft, about 5 minutes. Add salt, sugar, 3 cups flour, and butter. Stir, then beat with a spoon until well mixed.
2. To knead with a dough hook, beat on medium speed until dough is no longer sticky when touched and pulls cleanly from sides of bowl, 3 to 4 minutes; if sticky, add more flour, 1 tablespoon at a time.
To knead by hand, scrape onto a floured board and knead until smooth and elastic, 5 to 6 minutes, adding more flour as required to prevent sticking. Return dough to bowl.
3. Cover dough with plastic wrap and let it rise in a warm place until doubled in volume, about 1 hour. Uncover and beat with dough hook, or scrape dough onto a floured board and knead, to expel air.
4. With oiled hands, pinch dough into 24 equal balls (about 2 in. wide) and shape each into a smooth round. Set balls, smooth side up and about 2 inches apart, on 2 buttered baking sheets (14 by 17 in.). Cover balls lightly with plastic wrap and let rise in a warm place until slightly puffy, 15 to 20 minutes.
5. In a small bowl, beat egg to blend. Brush egg over tops and sides of rolls as each pan is ready to bake.
6. Bake in a 400° oven until rolls are well browned, 10 to 15 minutes; if using 1 oven, switch pan positions after 5 to 7 minutes. Transfer to racks; serve warm or cool.
Nutritional analysis per roll.