One of my plants apparently collapsed under its own weight, and the stem is clearly broken, where the plant stem meets the soil (a bit under it apparently). I think that if I move it and twist it for 30 seconds or so, I may be able to detach the whole plant. The plant looks healthy though. Do I have some chances of recovering it or should I just make pesto ?
I'm no expert, but if I really wanted to try to save it, I'd cut it back so it just has 5 to 10 leaves and prop it up with something.
I'm guessing that the plant got too tall from not enough sun. If so, when you're harvesting a small amount for a meal, consider nipping off tops - especially when it's trying to flower. This should make it branch more and keep the center of gravity lower, reducing the torque at the base. And in general, when just leaf-harvesting, do it near the top. There's more flavor there anyway.
Has this happened to more than one plant? If so, it might be cutworms, the larvae of the click beetle. They characteristically "cut" small, young, tender plants at their base, thus their name.
To control cutworms, you can put a little paper cylinder around your plants (or a yoghurt container with the bottom cut out, etc.), or you can run chickens through before planting.
Basil is very easy to root, just cut the stem clean and remove some of the bottom leaves, put it in a glass of water, or some vermiculite or perlite. Rooting hormone wouldn't hurt, but is not necessary. Then put it someplace that it wont dry out too much, greenhouse or window.
When you get roots plant them in some new soil away from whatever damaged it the first time, then harden it off to go back outside.