Using melted butter when making shortbread
If your recipe calls for the creaming method, then what you are doing, both with or without sugar, is harmful. 

Creaming the butter means to mix softened butter with crystal sugar, using a paddle mixer attachment or vigorously stirring with a spatula (warning, sore arms!). The sugar crystals beat tiny air bubbles into the butter, which make the shortbread rise later during backing. Neither cold butter nor melted butter work here. If you melt the butter first, your result will still be a shortbread, but will rise less and have a denser, heavier texture. Melting the butter first, then cooling it back and adding it is also bad; melting the butter breaks the emulsion and melted butter (which is actually an uneven mixture of butterfat and unstrained milk solids) behaves completely differently than "normal" butter. 

If your recipe already prescribes the muffin method, there is no need to mix the sugar with the butter, or to cool the melted butter at all. You mix the sugar with the dry ingredients, and add the butter together with the wet ingredients.