What do you do with a deep finger cut?

Stop the bleeding by direct pressure and seek medical attention as soon as possible. Let a medical professional evaluate the injury and determine if a suture or more will be necessary. Let the medical provider clean and dress the wound. A tetanus shot may also be in order. Some really nasty lacerations have been know to nick or even sever tendons, the latter of which will strip an individual of the ability to move a given joint.

Apply direct pressure until a pressure dressing can be applied. Seek medical attention.

Apply some kind of pressure dressing and keep it in place for several hours. Change it some time later being careful to remove it without opening the wound. Get in to the clinic to have it looked at.

These are just some options. The first one is arguably the best.