To the pan of your bread machine, add warm water, warm milk, egg, 1/3 cup of softened butter or margarine, sugar, salt, all-purpose flour, and bread flour. (Check your bread machine’s instruction manual and add ingredients in the order it recommends)
Use your finger to make a shallow well in the flour, then pour the active dry yeast into the well.
Make sure the bread pan is locked into place then select the Dough cycle on your bread machine. Close the machine and press “Start”.
Check the consistency of your dough during kneading and, if necessary, add additional bread flour 1/2 Tablespoon at a time until dough forms a slightly sticky ball that (mostly) pulls away from the sides of the bread pan.
Let bread machine complete the dough cycle and once dough has doubled in size, turn it out onto a lightly floured surface.
Sprinkle a little more flour on top of the dough and gently pat it in. Then very carefully flip the dough and lightly pat in the flour on that side too (to reduce stickiness).
Divide the dough into 16 equal pieces. Knead each one very lightly then pull edges into the middle, pinch together, and roll in your palm to make it round (see video for help). Place rolls on a cookie sheet lined with parchment paper, leaving space in between rolls.
Lightly spray plastic wrap with cooking spray and loosely cover the rolls (sprayed side down). Let dough rise in a warm place for at least 30 minutes, or until rolls have doubled in size.
Preheat oven to 350 degrees F. Carefully remove plastic wrap from rolls and bake for 12-15 minutes, until golden. Brush with half of the melted butter halfway through baking. Once rolls are done, remove from oven and brush with the remaining melted butter. Serve warm and enjoy!