I've seen this many times... I believe your issue is a failed needle/seat valve in the carburetor. This valve regulates gas flow into the float bowl. The needle tip is made of rubber and if it becomes at all distorted it will allow gas to flow into the carburetor and overflow it. The overflowing gas travels from the carburetor into the engine and cylinder head. It then slips past the piston rings and fills up the crankcase where the oil is with gas. That's why you smelled gas. This raises the level of oil way too high as it is diluted with gas to the point where basically a 50/50 oil gas mix is seeping into the combustion chamber and other places it doesn't belong when the engine is running. That immense amount of oil burning causes smoke. If you change the oil, voila, that gas is gone from the oil and it's back at the right level, so the smoke will stop. However, it will happen again if you don't address the carburetor. Either replace the needle/seat, replace the carburetor, or install an in-line fuel shut off valve that you shut off when you aren't using it. I'd start by using a shut-off valve and seeing if that solves the problem. Cheap, quick, easy and if that doesn't solve it no harm done. If it does work and if you're comfortable with it leave it that way. Otherwise, replace the carburetor or needle/seat.
This is probably happening because of the crappy ethanol gas we're forced to use nowadays combined with Briggs having awful QC as of late. And I imagine you may have let this thing sit with gas over the winter which can exacerbate those factors.