Mix together warm water, yeast and sugar in a small bowl. Let this mixture sit for 5 - 10 minutes. It will look foamy once the yeast has activated. This step is crucial for this recipe. If your mixture is not foamy, the yeast has not been activated.
Pour yeast mixture into the large bowl of a stand mixer. Add salt, flour, egg and butter. It's important to measure the flour by spooning it into the measuring cup because flour gets compacted when it sits in the bag or container. If you scoop the flour with the measuring cup it will stay compacted and you will add too much flour. Use a knife to level the flour.
Using a dough hook on a stand mixer, mix together for 3-5 minutes. The dough should be tacky and slightly sticky to touch. It should not be so sticky that it stays on your hands when you touch it. If the dough is too sticky add a pinch of flour. Only add one pinch at a time because adding too much flour will dry out the dough. Be sure to mix thoroughly between additions if you add more flour. If the dough is too dry, add a teaspoon of water and mix thoroughly. The end result should be a slightly sticky dough.
Once the dough has formed, roughly shape it into a ball and place in a large greased bowl. Cover with plastic wrap or a clean dishtowel. Let the dough rise for 1.5 hours or until doubled in size. This step is really important and is also known as "bulk fermentation" because it allows the yeast to continue to ferment. Fermentation helps with the structure and flavor of the dough.
Remove dough from the bowl and punch down to remove any air pockets. Divide dough into 12 equal pieces. Gently roll dough pieces into balls. Place dough balls in a lightly greased 9x13 baking pan. There should be 4 rows of 3 dough balls. Cover with a clean dishtowel and let rise for another hour. The dough balls should double in size.
Preheat oven to 350F. Bake rolls for 18-20 minutes. Melt the butter while the rolls are baking. Place butter in a small bowl and microwave for 20 seconds on high. Repeat in 10 second intervals until the butter is melted. I like to cover the bowl with a paper towel in case the butter splatters. Remove dinner rolls from oven. Use a pastry brush to evenly coat the rolls with the melted butter. Serve immediately.