Sarolea,
The flat bottom to the timing cover places it as 1930 or later; as does the just visible rotary breather valve on the crankshaft. The 1930 350cc models were mounted on a oil sump. As I do not see the required hole in the bottom of the timing chest for the oil pump drive shaft, the minimum date now moves up to the 1931 dry sump model, the A31. The A31 also lacked the bosses inside the timing chest (as does your engine) for the exhaust valve lifter mechanism. I am not sure just when, but these were reinstated on later engines. Possibly by 1934, certainly by 1935. Given there were likely no 1933 350cc machines manufactured (financial distress), that narrows the possibilities down to 1931-32, and a A31 or A32 model. The OHV lightweight based on the 350cc side-valve machines (K32 & M32) had a rounded shape to the top of the timing chest.
I do not know of any means to distinguish if it is a 1931 or '32 and I believe the model remained unchanged. This is reinforced by the fact the engine, frame, and gearbox prefix codes as assigned by the factory remained unchanged. (Exception, very late '32 may have received a new style plate cylinder heads, but your barrels are not for that type.) The engine prefix should be EV and will be stamped on the top of the crankcase. If you scrub the area indicated below you should find a trace of it followed by a three or possibly four digit number.
-Doug
