Revise that. The Hienkel is a four stroke and has ST30 (or ST40 at times, if you run to the book) in its gearbox/crankcase. Petrol in the 'gas' tank. So not much different to a Briggs and Stratton.
The engine will burn a surprising amount of oil if the mix is richened up a little. This is because ST30 is quite light. Indeed it used to be used in two strokes as lubricant prior to the adoption of specifically formulated two stroke additives. So an oily plug could mean a serious issue beyond a weak, or miss-timed spark. Even that is not a cure for heavy oil consumption, of course. It will only get worse.
The engine, being a single splash lubricated unit acts a bit like a pump as it strokes. It has no active breather valve, like the Isetta. So a worn bore (cylinder), rings or valve guides will see oil effectively being blown passed any resistance. The valve guides do not have seals on, a mod that can be made, but since the engines tend to run hot this needs to be good material, like an NSU.
A compression test will confirm a problem and it is then worth removing the head. Remember to take care not to loose nuts, washers and shims down the pushrod port, which is best stuffed up with a bit of rag. Once gone, never to return, unless your very lucky. Its Mr Engine rebuild time. On removal of the head you may find a build up of carbon holding the valve open, or a burned valve seat.
A positive compression test and high oil use will be valve guides. I believe these are available with valves at a reasonable price. They are fitted using heat and held in place with a long C snap ring.
Prior to all this, it sounds dumb, check you have the correct oil level on a correct dip stick. I have had two occasions where the engine was overfilled with oil. One an own goal by the collector, the other was a cut off dip stick (why?). In both cases oil was forced out where ever it could get out. One was a gnats whisker from hydrolocking. He had done the same thing to his Isetta, that was locking. Either that or there was miscommunication and his 'Conservator' had filled them assuming they were being static stored. Whatever, too much oil is not helpful when running.