First, turn on your oven light. Reason being: That will be your warm place to let your dough rise. Grease a large bowl with the oil and set aside. Line a 13"x9" pan with parchment paper. Set aside.
Melt butter in microwave on high about 25 seconds. Set aside.
Heat milk and sugar in a 2-cup glass measuring cup in the microwave on high for 1 1/2 minutes. Stir until sugar dissolves. Add water and melted butter. The temparature of this liquid mixture should be between 110°-115°F. (It is best to use a candy thermometer for this step. If you don't have one, make sure it's a little warmer than body temperature but not too hot to the touch or you might kill the yeast.) Pour into medium bowl. Sprinkle with yeast. Stir slightly, then let it sit for 5 minutes. You will see bubbles forming. This means that the yeast is alive and you can proceed to the next step. Note: If no bubbles form after 10-15 minutes, your yeast is no good and must be thrown out. Then start over with fresh yeast.
Meanwhile, in a large bowl, stir together 4 1/2 cups of the flour and the 2 teaspoons salt. (Note: Go ahead and measure out your remaining 1 1/4 cups of flour, set aside.) Check to see if your yeast mixture has "bloomed" (covered in bubbles). Add 1 egg to the yeast mixture. Stir to combine.
Create a well in the center of your flour mixture. Pour in your wet ingredients. Stir to combine with a spatula or wooden spoon. Your dough will be a little sticky still. Turn out onto a floured surface. Knead the dough for 8-10 minutes, adding as much of your remaining 1 1/4 cups of flour that you need. Knead dough into a ball and place in a greased bowl, turning once to cover the dough in oil. Cover with plastic wrap and let rise for 1 hour in a warm place such as your oven with the light on. The dough should double in size.
Punch down your risen dough, then knead for about a minute on a lightly floured surface. Shape into 24 balls. (You can weigh them if you want exact sizes.) Place on prepared pan. Loosely cover with plastic wrap or waxed paper. Put in a warm place again and let rise for 30-60 minutes or until doubled in size.
Preheat oven to 375°F (not the oven that has the rising rolls). Wait until they are risen, then heat your oven. Carefully brush the risen rolls with egg wash (simply a beaten egg). Bake in the preheated oven for 10-12 minutes or until the tops of the rolls are a light golden brown. Serve warm with butter. Rolls are best the first 24 hours. Or you can freeze them in an airtight container for several months.