Roast the Butternut Squash

1. Preheat your oven to 425 degrees Fahrenheit.
2. Cut the butternut squash in half lengthwise, scoop out the seeds, and rub the cut sides with olive oil and a pinch of salt. Place them face down on a lined sheet pan.
3. Roast for about 45 minutes until the flesh is completely tender when pierced with a knife and the edges are starting to caramelize. The kitchen will smell sweet and nutty. Remove from the oven and let them cool until you can handle them.

Make the Basil Oil

1. While the squash roasts, combine the fresh basil leaves and olive oil in a blender. Blend on high until completely smooth and the oil turns a vivid, bright green.
2. Strain the mixture through a fine mesh sieve lined with a clean tea towel or cheesecloth, pressing gently to extract all the oil. Discard the solids. The basil oil should be clear, intensely green, and smell powerfully of fresh basil. Set it aside.

Prepare the Gnocchi

1. Scoop the flesh from half of the roasted butternut squash into a bowl and mash it until completely smooth with no lumps remaining. Save the other half for another use or leftovers.
2. Add the all-purpose flour to the mashed squash and mix gently until a soft dough forms. You may need to add a little more flour or squash to get a smooth dough that you can roll out without sticking, but use a light hand — too much flour makes the gnocchi dense and heavy rather than pillowy.
3. Divide the dough into portions and roll each one into a rope about 1 inch thick on a lightly floured surface. Cut the ropes into 1-inch pieces.
4. Roll each piece gently over a gnocchi board or the tines of a fork to create ridges that will catch the sauce. The gnocchi should look like small, ridged pillows with a golden-orange color from the squash.
5. Bring a large pot of well-salted water to a boil. Drop the gnocchi in batches and cook until they float to the surface, which takes just a few minutes. Remove them with a slotted spoon and reserve about a cup of the starchy cooking water.

Make the Sauce and Assemble

1. Heat a generous drizzle of olive oil in a large pan over medium heat. Add the entire tin of anchovies and a pinch of red pepper flakes. Stir constantly as the anchovies dissolve completely into the oil, about 1 to 2 minutes. The mixture should smell deeply savory with no visible anchovy pieces remaining.
2. Add the green bean pieces and sauté until they are bright green, cooked through but still snappy, and lightly blistered in spots, about 4 to 5 minutes.
3. Add the lemon zest, season with salt and pepper, and toss to combine.
4. Gently add the cooked gnocchi to the pan along with a splash of the reserved pasta water. Toss carefully, letting the starchy water create a light, glossy sauce that coats every piece. The gnocchi should look golden and glistening.
5. Divide into bowls and finish with a generous drizzle of basil oil, freshly grated Parmesan, and extra basil leaves if you have them.