There have been 21 matches in which Broad, Moeen and Woakes all played, which I thought would be a good start.
stats.espncricinfo.com
Woakes made his ton in August 2018 though, so that narrows it down to just two matches, both of which Roy and Denly also played. We could probably a couple in that list that included nine players who would one day go on to become Test centurions, but that's not really in the spirit of the exercise, so I think we can rule out Woakes's ton as a way to get there.
It does seem to me that "Moeen played and batted down the order" would be a requirement so here's a list worth looking through:
stats.espncricinfo.com
This game has nine Test century makers in the lineup but Buttler hadn't made one yet at the time so it doesn't quite count:
Get cricket scorecard of 1st Investec Test, ENG vs AUS, Australia tour of England and Ireland 2015 at Sophia Gardens, Cardiff dated July 08 - 11, 2015.
www.espncricinfo.com
Buttler got dropped after that series but Hales and Taylor come in which makes it impossible again. Don't think it's happened any time recently.