Place the flour in a mixing bowl then gradually add the water. Use a fork to make a dough that feels soft but not sticky. Turn it out onto a floured pasta board and knead it until it is smooth and silky. This will take around 10 minutes.
Cover the dough with the bowl so it doesn’t dry out and leave it to rest for 30 minutes.
Pinch off a pea-sized piece and roll it outwards over the board with the palm of your hand to create a spindle shape. Pull your hand back diagonally across your body, pressing down gently but firmly on the pasta with the edge of your hand. You should create a twisted piece of pasta, which looks like a corkscrew. You can also try it with a bench scraper if you cannot get the hang of it with your hands.
Make the basil sauce by blitzing everything together in a blender until smooth. Taste for seasoning and adjust if necessary.
Bring a large pan of salted water to the boil and cook the trofie for about 2 minutes. The length of time will depend on how big your trofie are, so test one for doneness. Use a sieve or slotted spoon to scoop out the pasta once it’s cooked and place in a large serving bowl. Add the green beans, if using, to the hot water; blanch for 3 minutes and add to the pasta. Stir through the basil sauce. No extra cheese is needed.