How is a broken tendon in a finger healed?

Depends: Tendons tear, bones break. Your orthopedist will decide whether to splint the finger/hand and allow the tendon to heal in this manner or whether it needs to be surgically repaired.

Tendon healing: is much like wound healing but the tendon if unfer too much tension has gaps that weaken it, stitches take tension off the healing tendon to let it heal, too much healing and scar leads to reduced motion so limited motion helps to prevent adhesions. prtial tears less than 40% heal well on their own but beyond that suturing is needed