Edit: I don't recommend doing this as it affected greatly the response time of the system. I had to reverse the changes.
You change terminal with
Ctrl+Alt+2. Log as root and then do
pkill -KILL -u user
usermod --home /mnt/T/home user
Then remove the home folder and link it
sudo rm -rf /home/user
sudo ln -s /mnt/T/home /home/user
Alternatively you could move your home directory to a partition of your external drive and mount it at /home/user. This can be achieved automatically by adding an entry to the /etc/fstab file (assuming /dev/sdb1 is the partition of your external drive with an ext4 file system, adjust to your needs):
/dev/sdb1 /home/user ext4 defaults 0 1
After saving the file, either reboot or mount the file system using mount /home/user.
See fstab(5) for more mount options.
Edit: the first solution above ended up giving me some trouble. I ended up following this solution.