Millen and City parted ways on Monday morning after a run of just one win in 10 league outings, and with the Robins propping up the npower Championship table. The names of Dave Jones, Steve McClaren, Billy Davies and Alan Curbishley have already been linked, as speculation begins regarding the identity of Millen's successor.
But Sexstone has since told an assembled press corps that, although a speedy appointment would be beneficial, he and the Ashton Gate board are intent on bringing in the most suitable candidate. He said: "We're doing a thorough search and we're absolutely determined to get the right person."
He added: "We want a quick appointment, of course we do, but far more important than that is that we get the right person and we'll do that as diligently and as thoroughly as we can. I think the fans would expect nothing less than that.
"We'll be looking for someone with experience and with the ambition to take the club forward."
Sexstone was engaged in lengthy discussions with the City board, owner Steve Lansdown and Millen himself over the course of the weekend until an agreement was finally reached early on Monday.
Millen was then informed over the phone of the decision made by the Robins hierarchy, with Sexstone admitting the former boss was "devastated" with the outcome.
But the chairman insisted there were no rash choices made in haste during the process, going as far as revealing the board held differing opinions on the matter in hand. Ultimately, however, the thrashing at Bloomfield Road was the catalyst behind Millen's departure.
"Last week there was big disappointment at the way we lost the game at home and we were all devastated," said Sexstone.
"We all went to Blackpool really hoping we would turn the corner, and obviously to lose the way we did was very worrying for us in many senses. That really did set the alarm bells going. It's been a weekend of long discussions. It wasn't an immediate knee-jerk reaction, that's not the way we behave here at Bristol City."
Source: PA
Source: PA