In the bowl of a stand mixer, combine 1 cup of the flour, sugar, salt, and yeast. Add the butter and stir until thoroughly mixed, then stir in the water, followed by the egg. Stir well (you can do it by hand or with the paddle), and then add remaining flour 1/2 cup at a time until you have a soft dough.
Attach the dough hook to the mixer and begin kneading the dough. Dough should leave the sides of the bowl, but still stick to the bottom. Knead for about 10 minutes or until dough is smooth and elastic. Transfer to a well-oiled bowl and roll the dough around so that it's coated in oil. Cover bowl and let dough rise until doubled in bulk (45 minutes to an hour).
Punch down the dough.
Grease a 9x13 inch metal pan. Pinch off chunks of dough, gently shaping them into balls, and arrange dough balls in the pan spacing as evenly as you can. Cover and let rise again for about 45 minutes.
Bake at 375° for about 25 minutes or lightly browned. Remove from oven and brush with more melted butter.
If you end up doubling it you could probably get away with using just 1 packet of yeast rather than opening a new packet to make 3 teaspoons. Then again, if you measure your yeast out of a jar and don't use packets, that won't be an issue.