I will have to check my old catalog on Monday, but it looks very much like a Broadway faucet. That is not good news, because they are gone and no parts are available.
You mentioned speakman.. and that would not be out of line, because Speakman did make some internal stems and cartridges for the Broadway line. I still don't think you can find anything.
The outside shot.....Broadway was bought by Ingersol Rand ....http://company.ingersollrand.com/abo...s/History.aspx .....( see 1999 ). Being a standup company, when Ingersol Rand dissolved the product line around 2004 they did attempt to maintain service on the lifetime warranty to the original purchaser. It has been a couple of years since I have known anyone to use ths, but give Ingersol Rand a call.
Meanwhile, let's see if some of the other old-timers have a better recognition of your faucet.