One of they key ways flavor mellows is simply by the removal of yeast, with its own flavor and that of hop resins that adhere to it. The key is that yeast needs to have completed fermentation and metabolization of byproducts like diacetyl, which it does toward the end of fermentation.
At that point yeast has nothing more to do, and you're on to physical processes. Crash cooling the beer for a couple of days will start settling yeast, and also form chill haze. Now you can either filter the beer, fine the beer and wait a while for precipitation of yeast and chill haze particles, or just keep it cold and wait even longer for precipitation. Filtration doesn't speed up lagering, rather it replaces lagering to remove suspended solids.
If you like filtered beer, there's no reason for longer cold storage after the yeast is done and before filtering.
The typical method for lagers is to raise the temperature to 60°F or higher toward the end of fermentation to hasten the completion of fermentation, holding the beer warm until the yeast starts to settle on its own (showing it is no longer active.) If you dropped the temperature before fermentation was complete, just make sure you have in fact reached FG. In that case the yeast has also finished its work. Then you can go ahead and filter, carbonate and enjoy as soon as you like. As I say, once the beer is done, it's not ageing, just getting old. Filtration is one way to get it to your glass as fresh as possible.
I am one of those who has filtered for many years, but am getting away from it because I think it strips some flavor and body. But opinion is divided and a matter of personal preference. If you filter, take advantage of the time you can save!