We solved this problem in just 9 hours 37 minutes.
Here is how to do it.
Tools you need:
sudo apt-get install cryptsetup
sudo apt-get install dh-autoreconf
sudo apt-get install libcryptsetup-dev
1 - Dump the encryption header of your device using cryptsetup toolset, here /dev/sda1 was our device yours could change:
sudo cryptsetup luksHeaderBackup --header-backup-file headerOutput /dev/sda1
2 - Generate a dictionary with all the possible combinations of the part of the passphrase you forgot; We paid a professional solution for this but there are tons of tools and programming language to accomplish this task for free.
3 - Download and compile this free project (https://github.com/glv2/bruteforce-luks)
4 - Recover your password
./bruteforce-luks -t 6 -f dictionaryYouGenerated yourLUKSHeaderFile
Notice the step "4" will be extremely slow since it utilizes the CPU and the software is not optimized at all, that's why we bought a similar software made for GPU's and we also hired a powerful cloud distributed computing solution (I can't put names here)
In our case the cost of everything was exactly 4400 USD (developer + software + cloud service)
Lesson learned from now on we are storing the passphrase printed in a safe so this never happens again. Thank you and I hope this help someone in the same desperate situation.