In the bowl of a stand mixer, dissolve the yeast and sugar in the warm water. Let stand for 5 minutes to get foamy.
Mix in the butter, salt, and 3 cups of flour with a dough hook.
Once well combined, add the last cup of flour, ¼ cup at a time, until the dough forms a ball, pulls away from the sides of the bowl, and is only slightly sticky to the touch. YOU MAY NOT NEED THE ENTIRE LAST CUP OF FLOUR.
Knead for 3 minutes at medium speed.
Cover the bowl with a dish towel and allow the dough to rise until doubled. About 30-40 minutes.
Spray a baking sheet with non-stick cooking spray.
Gently dump the risen dough out onto a lightly floured surface. Roll out into a rectangular shape.
Lightly sprinkle the rolled-out dough with the garlic salt.
Sprinkle with the green onions, then the mozzarella cheese. Roll up the dough jelly roll style.
Slice the rolled-up dough into roughly 15 one-inch slices using dental floss or a super-sharp knife. I highly recommend the dental floss.
Place the roll slices on the prepared baking sheet. Cover the rolls and let rise for 17-20 minutes. They should nearly double in size.
Preheat the oven to 375 degrees. Bake the mozzarella green onion rolls for 20-23 minutes. The rolls should be slightly browned.