I haven't had the chance to play The Hanging Room, so please do take that into account.
That being said, first and foremost, the game is absolutely brimming with atmosphere. While Le Carré is explicitly called out and referenced (and especially the Tomas Alfredson film version of Tinker Tailor Soldier Spy), it is done in a way that gives the players a lot of leeway in defining the setting and specifics, and feels like a loving homage to the gritty, morally gray espionage stories that I love. The roles, relationships, vignettes and the ending are all incredibly well in genre, and I applaud the way this is all pulled off with a really good ear for what tone the narrative aims to be.
The framing of the narrative as flashbacks after the operation is brilliant. The way it forces the narrative to be tight and contained, the way it encourages conflicting viewpoints, hidden truths and outright lies and the way it ends, it is all incredibly in keeping with the best spy stories.
I also really like the idea of making connections on a board. As someone said, it would be super evocative to play using a cork board with pins and bits of red string...
Not having played the game, I feel very hesitant to say much about the rules. I really like the actions and everything seems geared to make for a good play experience. There were some things that didn't become clear to me on my first (admittedly, quick) read-through but which have already been commented on and clarified in the other comments. I know that for me personally, on my project, the time-frame and character limits of this competition both felt a bit restrictive this time around, and I think this game might also benefit (as I'm sure all games here would!) from a little clarification with regards to some rules. Examples of play would maybe also help, but of course, in the character limits given here weren't a possibility.
My gut feeling on a read-through is that some things in the rules could maybe be tightened up a bit and clarified, but it would take some playtesting for me to really see which parts this applies to.
So to conclude: This is an impressive piece of work, and I would love to see this game further developed. I also love the way you left yourselves the possibility of modular design with the operation types. Another possibility for further development could be to have different sets of settings that give some details about context, different factions etc.
All in all, a very strong entry, and as a fan of this specific type of spy story, I'm really glad to see people working with innovative games in the genre!