In a large mixing bowl, whisk together 1 and ¼ cup of flour and yeast.
In a medium saucepan, heat up milk, sugar, salt and butter. The mixture needs to be warm, not hot!
Using a hand mixer, mix the flour and yeast and wet ingredients, by adding the liquid slowly. Add the beaten egg and mix well. Mix the yeast mixture for 2 minutes, until it is smooth. It will be quite runny.
Switch to a wooden spoon and add stir in the remaining flour, ½ cup at a time, until the dough comes together and pulls away from the sides of the bowl. It will take a bit of stirring so be patient.
Place the dough onto a floured surface. Form into a smooth ball.
Wash the bowl you mixed the dough in. Add a tablespoon or two of olive oil and swirl to coat the bottom of the bowl. Place dough in bowl and turn over a few times to coat with oil.
Cover bowl with plastic wrap. Place in draft-free, warm place ( see note ). Let rise for 30 minutes. It should double in size.
Punch the dough down and place on floured surface. Divide into two parts and let rest for 10 minutes. In the meantime, butter a 13" x 9" dish.
Divide each piece of dough into 10 pieces and form into balls. See note below for tips on forming into smooth balls. Arrange in pan, in 5 rows of 4 balls each.
Brush tops of dough balls gently with melted butter ( see note ).
Preheat oven to 375 degrees F. Bake for 15 to 17 minutes OR until the tops are golden, the edges of rolls are white and they have risen to the top of the pan.
Cool before removing and serving.
Notes
If you forgot to pull out the butter from the fridge ahead of time, place it in a bowl and microwave for just 5 to 7 seconds or to just start melting it. Add to milk and sugar mixture. If you use really cold butter, the milk will get too hot before the butter melts completely and you will have to wait for the mixture to cool to use with flour or yeast. Hot liquid will kill the yeast and the dough won't rise.
If you don't want to use granulated sugar, you can use honey instead.
To form perfect rolls, form each piece of dough into a coin bag, by pinching the edges and forming the bag in your palm and then rolling the dough into a smooth ball.