Where in the Gemara do you find the concept of meat being "glatt"? Or where do you even find the concept of "kosher lmehadrin."
The overall concept is based on Chulin 37b:
(יחזקאל ד) ואומר אהה ה' אלהים הנה נפשי לא מטומאה ונבלה וטרפה לא אכלתי מנעורי ועד עתה ולא בא בפי בשר פגול הנה נפשי לא מטומאה שלא הרהרתי ביום לבא לידי טומאה בלילה ונבלה וטרפה לא אכלתי מנעורי שלא אכלתי בשר כוס כוס מעולם ולא בא בפי בשר פגול שלא אכלתי מבהמה שהורה בה חכם
Ezekiel Ch. 4 -- Ezekiel says, "Look, Almighty God, my soul is not contaminated; from my youth until now, I have not eaten an animal that was unslaughtered or severely injured; and rejected meat never entered my mouth." [Ezekiel wouldn't be bragging about keeping the basic law, he must be discussing measures of piety] ... "and rejected meat never entered my mouth" -- means I never ate from an animal that required a sage's ruling to allow it.
The Gemara is saying that if something was not obviously kosher, and took a great deal of erudition to allow it, then it would be a measure of piety to avoid it.
The Rama thus rules in Shulchan Aruch YD 116:7:
בהמה שהורה בה חכם מסברא, ולא נמצא הדין בפירוש שהיא מותרת, בעל נפש לא יאכל ממנה
An animal that required a sage's judgement to allow it, as the law was not explicit that it was permitted, an extra-spiritual person should not eat of it.
So that's basically "Glatt" as we use the term today, to mean "extra kosher." The Gemara says a Jew has to observe the milking, so at first glance, American grocery-store milk wouldn't be kosher ... but Rav Moshe Feinstein and Rav Yosef Eliyahu Henkin explain that "observe" means "ascertain" ... it required a sage's judgement to allow something non-obvious, thus it's a measure of piety to avoid it. Same with veal that was fed exclusively meat-and-milk products ... at first glance should be prohibited ... but, JD Bleich goes through the advanced concepts in Yoreh Deah about exactly how the formula is mixed, to allow it. If it took a lot of erudition to allow it, the pious would avoid it.
The original use of the word "Glatt" meant whether the lungs were obviously smooth, or if there were some questions about them that required an expert examination/consultation to allow. That would really just be another example of this concept.
And throughout the Shulchan Aruch, you'll find "In the case of XYZ, most authorities allowed it but some prohibited it, so an extra-spiritual person should be strict." (Or "if you're strict, bless you for doing so.")
As user160823 observed, the Gemara uses the term "Mehadrin" regarding how many chanukah candles to light.
In Hulin, Daf מו עמ' ב. The concept is perhaps not what you think it is. It is one of the Triefos, a hole in the lungs (called Sircha), which some Poskim (like Rashi) say that if you find some scar in the lung, you need to check if there is a hole, and others say that the scar indicates a hole no matter our further observations find, and the meat is Trief.
The concept of "kosher lmehadrin" is to eat only things with fewer Sfeikos (instances of doubt), Kulos (leniencies), and Machlokos (disagreements of Poskim),and is much wider than food. It is doing more then one needs to "by the book," and showing that we do Mitsvos not only because we need to,but we want to. It is found elsewhere, with regard to lighting Hanucha candles in שבת כה עמוד ב
The word glatt was invented in America when the union workers in the slaughterhouse industry were irreligious and unreliable, but couldn't be fired. So the Rabbis of the time ingeniously invented a new indefinable word which allowed them to choose the workers they pleased. This information is courtesy of Rabbi Belsky. This has nothing to do specifically with sirchos which the shulchan aruch and ramma argue about, although that might have been an area of distrust. Kosher limihadrin, or at least the theory of it, is a matter of not being somech on any hetteirim of bidieved and such, and bassar shehora bah chacham. Whether or not any hashgachos that make this claim live up to it, who knows. Marketing shpiel.