Sauté or steam the greens until wilted. They will shrink down to about ¼ cup once cooked. Purée the cooked greens and eggs in a blender just until combined and smooth, about 1 to 2 minutes.
For a beginner, you may want to use a large bowl to make the pasta. Otherwise, place the flour directly on your work surface. Make a large well in the center. Pour the egg mixture into the well. With a fork, incorporate the flour from the edges of the well into the egg mixture. Continue until you’ve formed a crumbly dough.
Knead the dough until it is smooth and elastic. (You likely won't need more flour on your work surface.) The dough should spring back after you make an indentation in it with your thumb. If it doesn’t spring back, keep kneading it. This can take about 10 minutes. Cover with a clean dish towel and let rest for 20 to 30 minutes.
Divide the dough into 6 equal portions.
IF USING A PASTA MACHINE: Dust each portion with flour before running it through the machine to flatten, starting at the lowest setting (0) and working up to a higher setting (5 or 6). Dust with flour between rollings as needed to prevent the pasta from sticking. Run the sheets through the fettuccine cutter.
IF SHAPING THE PASTA BY HAND: Lightly dust the work surface with flour, if necessary, as you roll out each piece of the dough to about ⅛-inch thick. Dust the dough with flour. Roll each piece of the dough up into a very loose tube. You will be slicing noodles from these tubes, so you don’t want them too tightly wound and stuck together. For fettuccine noodles, cut ¼-inch-wide slices from each roll.
Add the salt to a large pot of water and bring to a boil. Add the noodles and cook until tender, about 2 minutes. If you allow the pasta to dry, it will cook more slowly, in about 8 to 10 minutes. Drain the pasta in a colander.
Add the pesto to the now empty pot. Return the noodles to the pot and toss. Serve immediately.