The profession of the judge [+examples]

Solution 1:

Adjudication or judging.

To adjudicate: Make a formal judgement on a disputed matter.

(Oxford Dictionaries)

Solution 2:

The profession of a judge is Judge, like the profession of a policeman, is Policeman.

There are various titles for the different judicial offices, in different English speaking parts of the world: e.g. in England and Wales:-

In the Courts of England and Wales, Supreme Court, judges are called Justices of the Supreme Court...

Judges of the Court of Appeal, also called Lords Justice of Appeal, are referred to as "Lord Justice N" or "Lady Justice N."...

Lay magistrates are sometimes still addressed as "Your Worship"

and, in the US:

In many states throughout the United States, a judge is addressed as "Your Honor" or "Judge" when presiding over the court. "Judge" may be more commonly used by attorneys and staff... The judges of the Supreme Court of the United States, and the judges of the supreme courts of several US states and other countries are called "justices". Justices of the Supreme Court of the United States and Justices of other courts are addressed as "Justice (name)." The Chief Justice of the United States is formally addressed as "Mr. Chief Justice" but also may be identified and addressed as "Chief Justice (name)".

The justices of the supreme courts usually hold higher offices than any other judges in a jurisdiction, including a justice of the peace, a judge who holds police court in some jurisdictions and who may also try small claims and misdemeanors...

New York judges who deal with guardianships, trusts and estates are uniquely known as "surrogates."

A senior judge, in US practice, is a retired judge who handles selected cases for a governmental entity while in retirement, on a part-time basis.

Subordinate or inferior jurisdiction judges in US legal practice are sometimes called magistrates, although in the federal court of the United States, they are called magistrate judges. Subordinate judges in US legal practice who are appointed on a case-by-case basis, particularly in cases where a great deal of detailed and tedious evidence must be reviewed, are often called "masters" or "special masters" and have authority in a particular case often determined on a case by case basis.

Judges of courts of specialized jurisdiction (such as bankruptcy courts or juvenile courts) were sometimes known officially as "referees," but the use of this title is in decline. Judges sitting in courts of equity in common law systems (such as judges in the equity courts of Delaware) are called "chancellors."

Individuals with judicial responsibilities who report to an executive branch official, rather than being a part of the judiciary, are often called "administrative law judges" in US practice. They were previously known as hearing examiners...

Judges who derive their authority from a contractual agreement of the parties to a dispute, rather than a governmental body are called arbitrators. They typically do not receive the honorific forms of address nor do they bear the symbolic trappings of a publicly appointed judge. However, it is now common for many retired judges to serve as arbitrators, and they will often write their names as if they were still judges, with the parenthetical "(Ret.)" for "Retired."

...the United States legal system (like most Anglo-American legal systems) makes a clear distinction between professional judges and laypeople involved in deciding a case who are jurors who are part of a jury... Non-lawyer judges in the United States are often elected, and are typically either justices of the peace or part-time judges in rural limited jurisdiction courts. A non-lawyer judge typically has the same rights and responsibilities as a lawyer who is a judge holding the same office and is addressed in the same manner.

Source: Wikipedia

But for a catch all term for the profession, Judge works just fine.