Very interesting. Thank you!34288 is a 32 Hand Ejector, second model, first change, made in 1905.
Okay that date is a SWAG. The second model first change consisted of serial numbers 19426 to 51126, and were made from 1904 to 1906.
Your number looks to be about halfway through the run, so logic says the gun should be made in the second year of a three-year run. 1905.
The B off to the left of the serial number means that the gun was originally blue.