Digital Spy caught up with Shaun to discuss the project, how it felt to work alongside John Thaw's daughter Abigail and the new-found fame that comes with playing one of Britain's best-loved TV cops.
What is the setup for Endeavour?
"Basically, it's a story set in the '60s about a young police officer who returns to Oxford - the place where he went to university - to help solve a crime. There's a missing girl, and while he's there, he confronts a load of ghosts from his past.
"It's revealed that he got his heart broken in Oxford, failed to get his degree and left in a hurry, sort of shamefacedly. So what we get is someone who's confronting those issues, while also helping to solve the case of this missing girl."
That young officer grows up to be Inspector Morse - were you a Morse fan before signing up?
"I was aware of the character, but I'd hesitate to say that I was a fan. As soon as I got the call, I read the books and became a massive fan of them. It was only then that I began to watch the films.
"So although I'd seen them and I knew them, I hadn't really watched them, if you know what I mean. But as soon as I started reading the books, I was like, 'God, this is brilliant'. I started to wonder what kind of character they were going to make him as a youngster, who turned into this [familiar] character in his late fifties. I thought it could be really great."
How did you land the role of Morse?
"To be honest with you, I had a lot of good fortune. I'd done a job with the production company [behind Endeavour] previously and they called me directly about it. They asked me how I felt about playing it and what my ideas were.
"At that stage, I hadn't read a script and I didn't know much about it. Hence me doing so much research and reading the books before we took the meeting and before we began to speak at length about it.
"So it came about in a very straightforward way, really. Rather than reading a script and there being a bunch of people in the running for it, it was something that came my way. I was excited and picked it up and ran with it."
You have a scene with John Thaw's daughter Abigail in Endeavour...
"Yeah, that's right. She played a brilliant part of a newspaper journalist. She was fantastic. It was the first time we'd met. Apart from anything else, the work she was doing on set that day was brilliant.
"She was lots of fun, really light and worked in a way that was conducive to everyone else doing good work. There wasn't any kind of heaviness or sentimentality that she brought to set.
"It was fun - she was an actor coming to set to do a day's work, you know? And then we had the added bonus that her dad had played the part that I was playing a couple of years previous! But first and foremost, she was doing a great job."
Are you worried how fans might react to seeing someone other than John Thaw playing Morse?
"Well... you've got to suck it and see, haven't you? Check it out and if you don't like it, then turn over! But there's so many nods to stuff that's gone before, that I think it will please those fans.
"To be honest with you, my hope is that it brings a new audience to it. I don't think any of us wanted to be a slave to what had gone before. You want to... I'd hesitate to say 'reinvent', but to put your own stamp on it.
"Like I say, if people are going to be naysaying at the beginning, then that's fine, that's cool. But I hope people do enjoy it. I don't think too much about it or concern myself with it."
"I think, for me at least, he's someone who's lost and makes mistakes. He's very human, I think, and I think that's what differentiates him from a lot of the other detectives. He does makes mistakes, he's flawed, and it's really easy to see that.
"I think that's why people have previously latched onto him and enjoyed it so much. So it's my hope that people will continue to enjoy this interpretation of it."
Did you get a chance to meet Morse's creator Colin Dexter?
"I did, yeah - a bunch of times actually. He came to set a few times, and we met before we started filming. Colin has the final say on who plays Morse, and always has done. He was very specific about it, so we chatted a lot and got on well.
"He was kind of everything I wanted him to be - very funny, very witty and very intelligent. He was a great guy.
"He played a small part in Endeavour as well - he's kind of in the back of a shot. I know he's done that in all of the previous Morse films, so I was pleased that we honoured that."
14 million people watched the final episode of Inspector Morse - are you prepared for that level of fame?
"I didn't know that! I haven't really given it much though, to be honest, my friend. Like I say, I just hope that people enjoy what we've done. I've yet to see it myself, so it's a bit difficult to comment! But I just hope that people enjoy it."
Endeavour is a one-off for now, but would you be interesting in playing Morse again?
"Yeah, definitely. I had such a great time. Such an interesting time on it. I had a lot of fun, so if it was to come back around… but who knows? I want to see this and then I can be more accurate about it.
"But I certainly had a good time and I feel like, in terms of the story and where he's at, there's definitely room for growth. But I didn't want to think about that at the time. You can only do the task at hand, and I wanted to bring this film to life in a way that was vivid.
"If it goes well, brilliant. If it doesn't [continue], then it's still brilliant, because my hope is that we've made a great one-off. It honours what's gone before, but it's kind of something new as well. That is my intention - I just want this one to be great. And then I guess we'll see things in a new light next year…"
Endeavour airs on Monday, January 2 at 9pm on ITV1.