Add the water, egg, softened butter, sugar, salt, bread flour, and active dry yeast to the bread-machine pan in the order recommended by your machine manufacturer. A common order is liquids first, followed by butter, sugar, salt, flour, and yeast last.

Select the Dough cycle and start the machine. During the first 5–10 minutes of kneading, check the dough; it should become a soft, slightly tacky ball.

If the dough looks wet and loose, add flour 1 tablespoon at a time. If it seems stiff or dry, add water 1 teaspoon at a time, allowing the machine a moment to incorporate each adjustment.

When the dough cycle ends, turn the dough onto a lightly floured surface. Divide it into 12 equal pieces so the finished rolls bake at a similar rate.

Shape each piece into a smooth ball by gently pulling the edges underneath and pinching the seam closed. Arrange the balls in a greased 9-by-13-inch baking pan, leaving a little space between them.

Cover the pan and let the rolls rise in a warm, draft-free place for 30–45 minutes. They should look noticeably puffy and close to doubled in size before baking.

Bake at 350°F (175°C) for 12–15 minutes, watching for lightly golden tops. Avoid overbaking, since the rolls are at their softest when just cooked through and lightly colored.

Brush the hot rolls with melted butter as soon as they come out of the oven. Let them cool briefly before serving so the crumb can finish setting while the crust stays tender.