Hi Claus,
I have done some homework: E106 is a very interesting engine indeed.
It's possible (perhaps even likely) that this is a 348cc racing engine of the type used by Les Bailey, Tudor Thompson and others during 1920-21. It is identical in design to the 494cc S1 and 733cc S2 models that were available to the public, but I don't think the 350 was ever listed in a catalogue. A couple of years later we had in Australia 350cc "RA" and "TT" models, even though these were also not in the Douglas catalogues.
The 1914 OHV Douglases you mention were quite different. They had their cams down low - like the side-valve 350s - with very long push rods to operate the valves. Quite different to the later OHV engines with the full-width cam shaft above the crank shaft.
Clew mentions the 1920-21 348cc racers in his book (page 63 in my edition), and notes that Tudor Thompson broke World Records at Brooklands on 21 September 1921. 200, 300 and 400 miles at 64.68mph, 62.29mph and 61.00 mph, respectively. That's quick for a 350!
Clew has no photo, but here is one from The Vintage Years at Brooklands. I has no date, but is likely that September 1921 day.
On page 57 of Clew, there is a photo of Bailey (reg. no. AD5724, riding number 0) at Brooklands. Clew gives 1919 for the date. This sounds a bit early (perhaps someone can comment on the date of the registration number) - I'm not sure when these post-WW1 OHV bikes were first constructed.
Douglas engine numbers most often started at 101, so E106 might be the 6th 348cc OHV engine produced? Perhaps someone knows more.
Re the frame: I am 99% sure that ALL the OHV sports racing engines would have been housed in the frame we usually see for S1, S2, and even Bailey and Thompson at Brooklands. I think the frame that the engine is in at the moment is a standard 2 3/4 frame that someone has modified to accept the OHV engine. The top tubes have been lengthened, and the "cradle" added to clear the front cylinder.
Re CE10. In Australia, most of these engines would have been used for racing, even if they did start out on the road. The aluminium oil boxes on the rocker spindles are the usual "early" type (I see there is a 1920 patent date, and they were still used on early RAs, say up to 1924), and the return springs are present on many surviving S-model heads. It's a very sporty motor, but not necessarily a racing motor.
Perhaps Doug (or someone else) can comment on why it only has a 2-digit number.
Cheers
Leon