Step 1To make the pasta dough, place one-half cup flour and the salt into a large bowl or on a clean work surface and stir to blend the ingredients together. Make a well in the center of the flour and slowly pour in the beaten egg. Use a fork to stir the flour into the egg, then, as the dough forms, use your hands to bring it together into a ball.
Step 2Knead the pasta dough for several minutes on a floured work surface until the dough becomes soft and smooth. Form it into a ball and wrap in plastic wrap; allow the dough to rest for 30 minutes. 3. Divide the pasta dough in half, and roll out the first half into a long strip about one-fourth-inch thick. Repeat with the second half. Using the widest setting on a pasta machine, put the first piece of dough through once to form a long rectangle. Fold the short ends of the rectangle into the center and put the dough through twice more on that setting. Repeat with the second half. Lower the setting a notch and run both pieces through. Continue running the dough through the machine, each time at a lower setting, until each piece is as thin as a standard sheet of pasta, about one-sixteenth inch.
Step 3Change the attachment on the pasta machine to the fettuccine cutter and run each sheet of pasta through. Place the noodles on a clean, lightly-floured wooden board. Cut the noodles on the diagonal into 1 1/4-inch pieces. Let the cut pasta rest uncovered on the board while preparing the soup.
Step 4Saute the onion in the olive oil in a large pot until tender, about 5 minutes. Stir in the garlic and saute an additional minute or just until fragrant, but not browned.
Step 5Stir in the chicken broth and the grated tomato and bring the soup to a boil. Add the yellow beans, reduce the heat and simmer, covered, for 5 minutes. Add the lima beans and corn and continue to simmer 5 more minutes. Add the zucchini and simmer another 10 minutes, just until the vegetables are tender.
Step 6Season to taste with salt and pepper. Stir the pasta into the simmering broth. Cover and simmer about 2 minutes until the pasta is cooked through. Stir in the fresh oregano. Ladle into heated soup bowls.