No matter what I do, _my_ personal favorite shots are almost always the ones I drink after a considerable sit time, by which I mean, about the time it takes to become lukwarm, or 15-30 minutes or so. This begs the question, a couple actually.

1) What are the first components to go away during the sit time? By extension, I assume those components are objectionable to my tongue.
2) Am I unusual in this?

More background.
-I almost always drink my own home roast, so consistency is all over the map, though if anything, I mostly err on the side of too light, because I prefer bright, "third wavey" espressso and ristretto. It doesn't seem to matter if I'm using Sweet Maria's blends, or SO's, and too light or too dark (because most of my roasting has been done with a Rosto and no variac or temp probe). I now have a Behmor, but I have little time with it since Christmas, so my results are still a little scattered. No matter, my espresso tastes better after sitting.

-I'm one like Jimseven here: http://www.jimseven.com/2009/07/06/video-1-crema/ who doesn't like the taste of crema, though I realize it's a neccessary evil that comes when you have fresh beans, good equipment, and good technique. Obviously, crema is one of the things that goes away during the wait.

-I also have a slight tendency to mismanage my roasting schedule to where I often, but not always, end up drinking beans that have not enough post-roast rest. Still, it would be highly unusual if they did not have at least two days... but often not all the way to four days.

-Peter

I have also found that i enjoy my shots more after they cool for a few minutes. i don't know if it's temperature related or if some volatiles are lost...the shots seem smoother. I'm more along the lines of 2-4 minutes after pull ...not 15-30..they'd be room temp by then.

I don't even keep my cups on my cup warmer tray to avoid the shot staying hot.

It's similar for me, I see you are also home-roasting. I tend to drink my home-roasts way too soon, as soon as 5 minutes after roasting! Anyway, I try to roast enough so that I'm drinking most after 2 days or so, but usually I'm busy and my batches go faster, especially if they are really good.

I think letting the cup cool is very much an accepted technique, and one that mellows flavors out nicely. To me it seems to especially work better if I just roasted, reducing the baking soda like taste that comes up with too much CO2 dissolved from extra fresh beans. Pre-grinding and letting the grounds sit for a while can also help off-gassing which will artificially age a shot a bit more. All of this isn't quite as good as just waiting until the bean is at it's perfect say 7 day post-roast, but honestly I'm not good enough with my roasting schedule to be drinking perfectly aged beans all the time.

As a test, try to roast either a large batch, or a couple batches the same...then taste it everyday out to 14 days (even longer for some coffees like Yemen in my experience) and record some notes. This should give you the best day, it may be a lot longer than you think.

There's quite complex chemical processes going on both during roasting, as well as after roasting until the coffee is completely stale. I just think of it in terms of CO2 off-gassing, but there's really a lot more going on. In the end, the perfect way is to just be patient, but if you can't letting a shot sit, or pre-grinding can work some magic.

It really seems like most times, just as I hit a great day for that roast...I'm just on my last shot or two of beans.

When it comes to espresso, heat can definitely hide some flavors. I usually clean the group and such while the shot sits for a minute or two. For brewed coffee, I'm sure there is something similar going on but at a different scale. For instance, in a cooling cup I usually taste more acidity.

I let them cool as well too, unless just throwing in with ice and milk in mornings for an iced coffee. I don't even warm up the cups or flush hot water into them, just always taste better like that and letting sit for a few minutes. Drinking immediately the shots can seem to strong/sharp if that makes sense. Which is odd as your always taught/or read to keep cups in warming tray or flush group water and dry them and then pull shot.

I find smoother, fruitier notes emerge as a shot cools. Mouthfeel also improves as oils thicken as they cool. Chocolates and roasty notes tend to fade though, in my experience, although acids seem to persevere, which isn't always a good thing IMO. All in all it gets mellower but fruitier for me. I like it both ways, and love observing the shift, although I also love crema, which is why I prefer long blacks over americanos.

While I don't necessarily wait for my espresso to cool, unless I pulled it relatively hot, I have taken to stirring it to try to see if it makes any difference to the flavour. I've noticed recently that the last sip was mildly unpleasant - to be honest it was completely gross - so thought a stir may alleviate the problem, which it seems to have done.

As a general principle, both extreme heat and extreme cold mask flavor. To analogize, that's one reason why big beer companies like to tout how cold their beer is-if it warms up you can taste how gross it is more easily.

So why are we hearing out cups?

Oh, I was under the impression that most active forum users weren't paying any attention to the heat of their cups. You don't necessarily want the espresso to be cooling quickly as it's pouring. I still keep my cups on the machine, but I haven't pre-heated using hot water in years. I wouldn't say the cups are VERY HOT off the top of the machine with the way I use it. The machine is rarely on for very long prior to my shot(s).

I definitely find that a cool shot tastes nice. As a matter of fact, I just had a fairly cool shot from a cafe today and the shot was very delicious and matched the flavour description perfectly. Peach and chocolate!