Making espresso correctly - rerunning hot water several times through an espresso puck to fill an Americano

1. Most coffee shops use about 20 grams (18-24 grams in my experience) for a double shot of coffee, and 12-16 grams for a single shot. If you see a cafe using 20 grams for a single shot, its only because they are using double ristrettos (2 half-shots of espresso) instead of a single shot. Some people prefer the flavor profile of a double ristretto.

2. As you pull hot water through your shots, you extract different compounds from the coffee. Most people stop their shots when they've extracted all the tasty compounds, and I'll usually try to stop my shots before the coffee starts to taste bitter.

   If you prefer your coffee a bit more bitter and full-bodied, as well as with whatever other flavors you're extracting at the end feel free to pull extra shots from the same coffee, however I would normally reccomend just adding extra shots to the americano instead.

   All that being said, Barista Champion Matt Perger reccomended during his world barista run a process similar to what you're doing. He reccomended using a coarser grind and running the shots for way longer to achieve a coffee similar to a filter coffee or americano. However he also has the benefit of expensive professional machines that are highly tuned, so your experience may vary.

Summary

My reccomendation if you are going to pull the shots more than once is to use a coarser grind for your coffee. Otherwise feel free to pull the shots as long as you like until you dont like the taste. You could also try pulling the separate shots into separate cups to taste them before mixing them together.

Expanding upon leon's answer:

I learned to make espresso over about six months working at a bakery/coffee shop that used a massive Italian machine with a gigantic brass (or at least, brass-coated) pressure tank. I couldn't tell you for sure how much coffee or water we used because I don't remember, but we always pulled double-shots and the goal was a 23 second pull, plus or minus 2 or 3 seconds. After that point, we had a further 30 seconds to mix in everything else (steamed milk, mostly, but we made the odd Americano) before the espresso went bitter; I don't recall being told why. It has been a few years, but here's what I infer from that experience.

The speed of your progression through the compounds extracted from your coffee (from light and floral at the beginning to bitter and acidic at the end) is a function of the time your coffee spends immersed in water roughly divided by the coarseness of your grind. So a finer grind is likely to become bitter fairly quickly, but will be strong after mere seconds of immersion. This is part of what makes espresso different from pot coffee -- finer grind and shorter immersion times.

If you can control the volume of water used, then the amount of time your espresso is immersed in hot water is a function of water pressure roughly divided by tamp. If you tamp really hard, your shot will run really long and come out looking and tasting roughly like tar. If you don't tamp at all, it takes 5 or 10 seconds and is fairly watery. Note that even if you're using a machine that doesn't pressurize, your water pressure isn't zero, but whatever the surrounding atmospheric pressure and gravity provide; usually this isn't much.

Therefore, if you're planning to immerse your coffee for a long time, in this case by continuing to run water through it to fill your Americano, there are two things you can adjust to get the flavor the way you want it. One is by making your grind coarse, and the other is by reducing the weight of your tamp; the results of each of these actions will be slightly different, but both seem worth testing.

Something else that may be worth considering is that coffee will still extract in room-temperature or even cold water; this is the principle behind cold-brew coffee. I've done it, and liked it, but my understanding is that it changes slightly the order of extraction, and even eliminates some compounds from the process. This, coupled with the temperature difference, may change the taste and mouth-feel beyond what you're comfortable with.

To find more modern resources, I googled "Tamping Espresso". Mostly what I found were pages encouraging a 30lb tamp, which is way more than we ever did, and whether it was proper or not, people did tend to like our espresso. Maybe we ground extra fine.

Anyway, hope that helps explain a little. Let us know if you arrive at a final process that you like.