Note that some commercial breweries use large yeast starters and finish fermentation in 3-4 days, they then force carb and bottle for grain to glass time less than 10 days.
You can do the same on the homebrew scale if you are inclined to do so with some styles.
There really isn't any advantage to naturally carbonate a beer that I know of.
But there are advantages to allowing yeast to maintain contact for 2 weeks or longer. Yeast can be messy eaters, more time allows them to clean up after themselves reducing off flavors. They also do a good job of scrubbing oxygen. Full attenuation takes additional time with some yeast strains. High Gravity brews also take more time.
The best grain to glass times will come from low gravity beers, something like a cream ale. Brew something up at 1.05 or lower and used a good yeast starter, many calculators are available to help you size your starter properly.
Lag phase should be near completion with your starter and yeast will go to work in a few hours. Keep temperature in the proper range, most ale yeast do well at 68F. Fermentation will be vigorous so use a blow off. On day 2-5 take gravity readings; needs to be the same for at least 2 days before racking. Cold crash fermentor and rack off of yeast; force carb by the "rolling" method. Enjoy.
