When performing procedures on the hands and upper extremities, many options are available for anesthesia. Regional anesthetics have a unique application for this kind of procedures, the most frequently used is the digital block, where you block nerve branches by applying anesthetic at both sides of the finger also called Oberst block.
That's both deep and clean for a utility knife. Usually deeper wounds are from dull knives because they are easier to lose control of, while sharp knives are easier to control.