Neat little bottle, evenw ith the sheared top. I'm not sure on an exact date, but I can give an estimated range. The patent number on the bottom (2035404) was applied for in April 1934 by John J. Quinn, and granted in 1936. The bottle has an Hazel Atlas Company mark on the bottom; the company was sold in 1964, but that makers mark continued to be used for a few years by Brockway as it was more expensive to change the plates to erase the mark than just keep the old makers mark.
I have one Pinaud. Much smaller than your's and mine has a crimped on metal cap with a centered narrow pump(?) hole. I looked it up and noticed that just about all the Pinaud perfume bottles are bringing a fairly good price, even if the condition isn't 100%.--Welcome to the wonderful world of bottle clean up + restoration.