here. Poul Anderson's Psychotechnic History does include one story specifically about the Psychotechnic Institute.
The first four stories, collected as The Psychotechnic League (New York, 1981), are:
"The Sensitive Man";
"The Big Rain."
"Marius" introduces Valti (off-stage), Fourre and Rostomily. In "Un-Man," Fourre runs the UN Secret Service whose agents, the Un-Men, include a "Brotherhood" of Rostomily clones. Another Un-Man, not a Brother, stars in "The Big Rain." Valti's work is continued by the Psychotechnic Institute and "The Sensitive Man" is about two members of that Institute. Thus, the second, third and fourth stories follow, directly or indirectly, from the first.
However, the Institute works on two levels. It tries to change individuals and society:
"'The Institute is trying to do two things - create such a citizenry [intelligent, alert and tough enough to defend freedom against power] and simultaneously to build up a society which itself produces men of that kind and reinforces those traits in them. It can be done, given time.'" (p. 173)
The story dramatizes one changed individual, the title character, but otherwise, like other installments of the series, discusses changing society in the abstract. This is why I had not remembered any particular story as specifically focusing on the Institute. The Sensitive Man owes his enhanced powers to Tighe System training which can be discussed apart from the rest of the Institute's work.
In the later story, "Brake," a single character is a member of the Order of Planetary Engineers and of the Rostomily Brotherhood and has Tighe System training but is not a psychotechnician.