In the bowl of an electric mixer, combine the yeast and warm water. Add the honey, oil, salt, and egg and mix well. Add 3 cups of the flour and mix until the dough comes together in a sticky mass. With the mixer on low speed, incorporate the remaining ½ to 1 cup of flour a few tablespoons at a time. Continue kneading on low speed for about 8 minutes, until the dough is smooth and elastic.
Transfer to a lightly oiled bowl, turn once to coat, and cover with plastic wrap. Let rise in a warm, draft-free spot until doubled in bulk, about 2 hours.
Turn the dough out onto a lightly floured work surface and knead for 30 seconds. Cover with a towel and let rest for 10 minutes. Punch the dough down and divide into 10-12 equal size pieces. Shape each piece into a smooth ball and place into a round, lightly greased 9- or 10-inch round baking dish, spacing evenly.* Cover and let rise in a warm, draft-free spot for 20-30 minutes.
Preheat the oven to 400˚ F. Mix together the melted butter and honey, and brush the tops of the rolls with the mixture. Sprinkle lightly with coarse salt. Bake for 15-20 minutes until the tops are golden brown. Cover loosely with foil and continue to bake about 10 minutes more, or until the the center of the rolls registers 190˚ F on an instant-read thermometer. Let cool slightly before serving.
*To make the rolls a day in advance, prepare the dough as above through to shaping. Cover with plastic wrap and transfer to the freezer. Leave there to stop the rise, 2-4 hours. Transfer the baking dish to the refrigerator and leave overnight. The rolls will begin to rise again slowly while refrigerated and will eventually puff up and into each other. The following day, about 18-24 hours after the dough was prepared, bake as directed above. You may need to add 2-3 minutes to the baking time, but monitoring the internal temperature is the most reliable way to check for doneness.