Hi all, thanks for the replies, they're very helpful. I sampled too much of this evenings experiments and consequently I'm up far later than I should be. (And now that I've written it, this post is way too long; it's age, I tells ya; take my advice, stay
thirty-something!) The downside is that I'm going to regret it when I have to get up in four hours. The upside is I think I just realized what my problem is.
I think this has been yet another lesson in "Know your equipment".
akallio's description of a shot on the Duetto is exactly what I'm seeing, and that made me realize I'd made a bad assumption: I assumed GriffDeLaGriff's machine, because it was an E61 like mine, was a rotary.
Ok, now that you've had a jolly good laugh, all I can say is, I hope the coffee that went up your nose was very, very hot.

I just googled the Cuadra and -- correct me if I'm wrong -- it's a vibe. I should have known better too, having had a Brewtus III-V for about 3 weeks a couple of months ago. After I realized this I remembered that, aside from being quieter, the coffee showed up sooner on the Duetto as compared to the Brewtus and Silvia. So the 6-12 second makes perfect sense now that I remember how my shots came out back then.
Jim, thank you for the 101 -- I'll try those adjustments to fine tune my shots.
geoffbeier, I wasn't trying to chase down a taste defect per se, just looking to improve my knowledge and skill, with the goal of being able to consistently produce a good tasting shot. I've learned quite a bit since I started brewing espresso myself and I'm trying to stay open to other ideas, especially as my ability to taste the coffee evolves.
GriffDeLaGriff, I'm not sure why I went with changing dose instead of grind. Sometimes I vary grind while keeping dose constant, sometimes the reverse. For example, I ran through my
Caffe Fresco coffee and my next batch hasn't arrived yet, so tonight I picked up 1/2 a pound each of Brazil Cerrado and Sumatra Estate (both supposedly 4 days old) so that I had some coffee in the meantime. (I've never had these coffees before either, and knew nothing about how they'd come out as espresso, so I was boldly going where I had not gone before. Or foolishly wandering where angels fear to tread; not sure.)
Anyway, continuing the experiment of trying to get beading to happen at around the 8 second mark, I started with the Brazilian at 15 grams. I think it took around 10 seconds for first beads. I stopped it around 45 seconds or so. I had about an ounce of overextracted coffee. Not too good, but I've done worse. Since I've never had this coffee before, I wasn't sure what to expect. Next shot, 2 notches coarser on the grinder. A little faster, but still over extracted. Next shot, one notch coarser and I upped the dose to 16 grams, just for the giggles. A little better but still very slow and overextracted; it too wasn't worth drinking. Enough for now.
About an hour later I went back to the machine to play with the Sumatra. As a starting point, I kept the dose at 16 grams as well as the grinder setting of the last shot. Right away I could tell these beans were going to be different. Whereas 16 grams of the Brazillian didn't fill the basket (before tamping), the same weight of the Sumatra beans almost overflowed it.
Whoosh! Beading happened around 2 seconds and I stopped the shot at 12 since it already filled the glass. And gasey, oy! Ok, way underextracted... but I tasted it anyway.

Surprisingly, it wasn't that bad! I decided to try four notches finer on the grinder: beading around the 3 second mark, and 20 seconds or so later I stopped the shot. Tons of cinnamon-colored crema, tasty, but not quite there. I think it was around now that I abandoned the idea of getting it to bead much later and just went with what was working. One more notch finer: I think I stopped it around 27 seconds, and the crema was cinnamon-colored with some dark flecks. This was going to be my last shot of the night so it went into a latte, since I'm also practicing my steaming. However, I tasted it before adding the milk and it was good. The latte had a warm, maybe milk chocolate taste to it. Not quite the same kind of chocolate as the Luna Nuova but good all the same.
In the morning I'll try that last dose/grind again and drink it straight. In the evening I might go back to Brazilian and start grinding coarser and see what I can do with that.
Thanks again for helping a newbie find his way. :p