In your quote there are two parts. First, there is the concept that one who commits a crime shall be sentenced or punished, specifically by execution. And two, that same person who determines the sentence is to be execution, must be the one who carries out the punishment. The first concept has existed since basically always. The second is not as clear. However, it seems the historical precedence is that the establishment of the first concept negated the second.
It should be noted here that the quote is by George R.R. Martin and is widely seen as an argument against the death penalty in modern day society. The idea is that if you have to kill a person yourself, you're less likely to sentence them to death because it forces accountability and personal responsibility. Whether or not the death penalty should exist is a wholly separate discussion and will not be addressed here.
The first concept
The first concept of the quote is saying a man can be put to death for committing a crime. To discuss this idea we look to the theory of punishment, or why we even punish people for crimes in the first place. Is it to deter others from committing the same crime? Is it in the hopes of rehabilitating them or having them learn their lesson? Is it to incapacitate someone so they cannot commit the same or other crime again? Or is it some sort of retribution for the harm done? All these justifications factor in to the particular sentence for a crime.
Prior to sentencing or punishment, or really just prior to any semblance of a society or criminal justice system, humans killed other humans typically out of retaliation or vengeance. This included land disputes, tribal warfare, blood feuds, etc. When wronged, people took matters into their own hands. As a result, often times retribution was worse than the original crime committed. As communities and early civilizations grew and organized themselves, bodies (either of law or of religion) were established whose purpose was to ensure order and law. Unchecked retribution and personal revenge was not an effective way for your civilians to live. Therefore, these bodies were to be the only ones who could enact retaliation and or punishment for a crime against the community or persons in the community. This ensured that the enacted punishment was the only punishment.
The overwhelming precedent set by the history of these bodies is that a person should suffer in proportional response to the crime they commit. Punishment should not be personal. It is directed only at the wrongdoing itself not the offender, and has inherent limits. The beginnings of this theory are found in the principle of Lex Talionis, or "an eye for an eye".
An eye for an eye is often seen as an argument for justice and retribution but it should be noted that the concept actually was intended to be a limit on punishment. It is better read as only an eye for an eye. As stated above, to avoid unnecessary retaliation, there would be objective standards of punishment for crimes and those punishments would be spelled out by the laws. Already, it seems as though the idea that one man should pass judgment and decide the sentence of punishment for a crime negated the idea that the same person should do the execution themselves.
Since those bodies began punishing crime, as opposed to individuals seeking retaliation, "Capital Punishment", or state sanctioned execution, has been supported. The Code of Hammurabi, one of the very first legal texts from human history codified the death penalty for 25 crimes. The Hittite Code and Draconian Code also made death the punishment for certain crimes, as did the Twelve Tablets of Roman Law.
Religious writings also have weighed in on killing and or the death penalty repeatedly, as seen in the Bible, in the Talmud, and in the Quran. Philosophers have taken their swings at the same concept, including Cicero, Kant, and Hegel among others. Even classic literature has weighed in. Chaucer's Canterbury Tales includes The Second Nun and The Nun Priest's Tale. Within this story we find the famous quote "Murder will out". While today many people use this quote to argue that wrongdoing will always be discovered and no bad deed goes unpunished, the original meaning in the story is different. The original wording was "Mordre wol out, that se we day by day./ Mordre is so wlatsom and abhomynable/ To God that is so just and reasonable, / That he ne wol nat suffre it heled be, / Though it abyde a yeer, or two, or thre." Chaucer is arguing within a humorous story that society must have a desire to punish murder, however there is a difference between illicit murder by man and killing for justice or what God would determine to be just. In fact, Mordre comes from the french word "mordre" meaning "bite" or the compensation one must pay for their crime.
So, whether or not people can be put to death as punishment for crimes they commit has historical precedence basically everywhere you look. However, the second concept of the quote, that the person who sentences someone to death should be the one to carry out the execution, is much less clear.
The second concept
Death as a punishment serves two purposes. First, it is an attempt to dissuade or deter others from committing the same crime. And second, it is retribution for the act committed.
Because the main goal in many ancient societies was deterrence, often early punishments for death were methods of torture that ultimately ended with death. For example:
*Crucifixion was seen as a just death because it was slow and painful. The term excruciating literally meaning "out of crucifying".
*"Broken on the wheel" was a form of punishment where murderers and robbers would be have their bones slowly broken and beaten until death.
*Traitors in England were drawn and quartered (literally chopped into 4 pieces) as it was seen as an effective deterrent to other would be traitors.
*People were burned at the stake or boiled in hot liquid as it was thought fire / heat were symbolic of hell.
*One Assyrian tradition was flaying, where ones skin is removed painfully and methodically, and was thought to be done because the ruling class was superstitious about the physical body of person.
In these instances it was not really the person sentencing conducting the execution because the execution was drawn out or not done by the hand of any one person.
Other ancient civilizations focused less on deterrence and more on retribution making the punishment some sort of symbolic representation of the crime.
*In ancient greek and roman militaries, soldiers found guilty of desertion, cowardice, stealing, or false witness (among others) were have thought to violate the trust of their fellow soldiers. Therefore their punishment came at the hands of their fellow soldiers. Two such punishments are fustuarium (a solider was forced to be cudgeled to death by his fellow soliders) and "running the gauntlet" (yes this where the idiom comes from).
*Forced ingestion of poison was often a punishment for those who spread lies or information the ruling class did not appreciate (think Socrates). The concept here that your lips uttered false witness so you should be forced to drink your death.
These instances are somewhat closer but still fail to really establish precedent for the person sentencing also being the executioner.
Interestingly, the concept of an independent executioner is fairly modern. It was a profession first described in the middle ages. Executioners were prescribed by the law of the city/state they worked in. It was usually recognized as a family trade, with title passing from eldest son to eldest son, or nephew. The laws of the locality dictated where they lived and who they could touch or speak to.
The reasoning for all of this rigidity was that executions in the middle ages had a distinct purpose. You were put to death as punishment for a crime against the state or a crime against God. Your death was justice and should your execution be botched or mishandled, it could be seen as divine intervention or an affront to the authority of the state to punish. So the State wanted to make very sure that the person who was killing all of the punished was really good at it. In fact, according to one book, in some German towns to make sure executions went smoothly an executioner was permitted three strikes (like actual physical strikes) before being grabbed by the crowd and forced to die in place of the sinner.
This almost begs the question, well what about before executioners were a thing? There surely must have been some small self-ruling societies that had a person both sentence and punish the accused? The honest answer is we just don't really know. It's entirely possible but for the most part, since the beginning of criminal justice or the origins of sentencing and punishing people for a crime, the actual act of execution was not performed by the person sentencing the crime.
For further reading, you can check out the various legal codes, Canterbury Tales, Kant's Science of Right, Hegel's Philosophy of Right, or the religious texts I mentioned but some others have done a good job going into them as well.
"The meaning of the bible:" Knight, Douglas A. and Levine, Amy-Jill (2012).
"Religious Beliefs as a Basis for Ethical Decision Making" Bond Kenneth, Ph.D (1998).
"The Torah — A Modern Commentary" Plaut (1981).
"Medieval and Early Modern Murder: Legal, Literary and Historical Contexts, ed." Tracy, Larissa (2018).
"Eye for an Eye" Miller, William Ian (2007).
"A short history of the Executioner" Edwards, Stassa (2014).
"The Faithful Executioner" Harrington, Joel F. (2013).
"The Executioners Who Inherited Their Jobs" Vatomsky, Sonya (2018).