Combine 3 cups of flour, yeast, sugar, salt, warm milk, butter and egg in stand mixer bowl. Attach dough hook and turn mixer on lowest speed and mix until flour is incorporated, scraping down sides of bowl as necessary. Increase speed to medium and blend for 2 min.
Add 1⁄2 cup more flour and blend with dough hook until incorporated. Add another 1⁄2 cup flour and repeat, mix at medium speed for another 2 min. until a ball of dough is formed. Add additional flour as necessary. Dough should be slightly sticky and soft, and pull away from edge of bowl. Transfer dough to a lightly greased bowl and cover with a towel or plastic wrap. Let rise for 30 min. at room temperature.
Remove towel or plastic wrap and deflate dough by punching down lightly. Pinch off pieces of dough and form 24 rolls. You can weigh them to keep rolls close to the same size. Mine were about 2 oz. each, but this will vary depending on how much flour you added.
Transfer rolls to a lightly greased quarter baking sheet or 9×13 baking dish. Cover with towel or plastic wrap and let rise for an additional 30 min. at room temperature. Preheat oven to 375ºF. Bake rolls for 12-15 min. or until golden brown and cooked through. If rolls are getting too brown, just tent rolls with foil.
Remove rolls and brush hot rolls with melted butter. Serve immediately or store cooled rolls in a plastic bag for up to 3 days.