I have been considering the same for a while and wondered whether the solution is having regular micro-events, rather than a "serious day", with all the negative implication this has for quality on other days.
And by this, I mean one or two pre-selected missions (perhaps from a list so that there is flexibility in size) that are known to exist and work, played on a set time on a set day. No real planning involved from an event perspective beyond selecting the missions a week in advance and making a announcement and perhaps an "expression of interest poll". No need for complicated sign-up / slotting arrangements, selection of leaders, pre-session briefing etc just a fixed thing that people can rally around and foster a 'serious' but supportive atmosphere. This could even work as the start of a weekend play session to set the tone for the rest of the night.
Obviously players on the server can then go on to play other missions outside of the ones selected for the 'micro-event', but hopefully with a better base attitude from those on the server.