I have a Ride on mower with a 12.5 Briggs & Stratton engine. When mowing on the more sloping parts it starts blowing Blue/white smoke. It's a great little unit that cuts way above its weight, BUT, is there an engine design capable of handling a sloping site better?

Hi,
First thing I would try would be an oil change; old oil gets more easily consumed. And do go cheap on the oil; a high quality name brand 30wt oil of SG or SH rating is called for. That rating will be printed on the label.
Peace,
Clarke
PS 40 to 48 fl. oz. depending on the sump pan, just go slow toward the end in filling.