That engine was made on 1 June 2009, in one of Briggs' plants at Wauwatosa, Wisconsin. Wauwatosa is the location of Briggs' head office.
Here is the Operator's Manual:
http://bsintek.basco.com/BriggsDocumentDisplay/iorpzNJU0HCjhJvcv4.pdf

Here is the Illustrated Parts List:
http://bsintek.basco.com/BriggsDocumentDisplay/18kvuIXSnfBhU7y.pdf

The engine is called 450 Classic, meaning it is a mildly updated 1960s design. It is interesting that it was made in the US as late as 2009 - AFAIK, all Briggs side valve engines are now made in the Chongqing plant in China. The last to move were the large engines, which were scheduled to be relocated late last year.

I'll add another possibility to bigted's list. When this mower was running grossly overfilled with oil, the whole crankcase breather system would have been completely full of oil and slugging large amounts of it through the breather pipe into the carburetor's air intake. Draining the excess oil from the crankcase did not drain the breather system, which is part of the reason it continued to smoke for some time. (The other reason is that the muffler was full of liquid oil, and it had to cook this to carbon then burn it out.) There is probably a fair amount of oil still in the valve chest and the crankcase ventilation valve may be only working part of the time. There may also be quite a bit of carbon in the muffler. If that is the problem, it will probably sort itself out over a couple of hours' running.