Chosen Solution

I have a 3GS that has the digitizer connector damaged (2 pins at the very top have been broken off). As a consequence, any new screen put on the 3gs will fail to recognizer any touch. My main need is to extract the data out of the phone. I heard transferring flash chips is next to impossible, but the display connectors are attached to the logic board anyway. Is there any way to fix this? http://i.imgur.com/cBktlLE.png The picture is flipped upside down, so the missing pins are on the bottom

Yes it is possible to replace this. Hot air station to desolder the old one, wick the old solder, and drag solder/pin by pin solder a new one on (using flux of course). Connectors are available for these usually, or you can desolder one from a donor board by heating the other side of the board (to avoid melting the plastic on the connector)

Yes it is possible but for that you need micro-soldering kit. secondly it is difficult to find display connector only for your iphone 3gs. If you manage display connector from electronics part shop that it is fine. if you not able to find connector then better to find any dead iphone3gs and desolder display connector from their. So you can replace display connector.