Does TL have a comparable feature to Avaya's Twinning feature?
Yes, you can configure find me follow me in a way that it will ring extension and any other phone - including mobile.
Third Lane Technologies
Multi Tenant Asterisk PBX
But the other feature twinning provides is the functionality when the user picks up the extension, the call is resumed on the phone and disconnected on the mobile.
Is that configurable some how?
technically that's not exactly whats going on, but in theory, its possible. Usually that feature invokes DTMF to perform a transfer of the call back to your desk phone. Currently this does not exist. It would require creating a special application in features.conf to find a stored value in the channel variables and then perform a blind transfer to that device. The work-around people used was adding the Tt option in the Dial application arguments, but that leaves your phone system vulnerable for some pretty bad call-redirection that's exploited by fraudulent telemarketing scams.
Thirdlane/Asterisk Support available