Combine 1 1/2 cups flour and 1 teaspoon of the salt in a large bowl. Make a well in the center and add 3 beaten eggs. Slowly incorporate flour into eggs from the sides of the well with a fork until a thick enough mass forms that you can dump the remaining loose flour and dough onto a clean work surface.
Push loose flour to one side of the counter and continue kneading the dough until it feels soft and elastic adding back the extra flour from the bowl when it starts getting sticky. Try not to add extra flour if possible. Once a smooth dough forms, knead for 3 minutes more. Place dough in a plastic bag and let rest at least 30 minutes or up to 3 hours.
Divide dough into 4 equal pieces. Leave one piece out and return the rest to the plastic bag. Using the least amount of flour possible, roll out the dough, stretching and turning as you go, until it is 1/8-inch thick.
With a knife, cut pasta sheets into 1/2-inch wide-by-2-inch long strips. I also use the rounded end pieces but you could cut them off if you want perfectly rectangular noodles. Place strips in a on a parchment-lined baking sheet in as thin of a layer as possible and set aside. Repeat with remaining pieces of dough. The noodles can be made up to 3 hours in advance, leave uncovered at room temperature to dry.
Heat 3 tablespoons of olive oil in a large soup pot over medium-high heat. Add 1 diced onion and season with salt and pepper. Cook, stirring occasionally, until onions start to become tender. Add 4 cloves choppedd garlic, 2 sliced carrots, and 2 stalks sliced celery and seasons again with salt and pepper. Cook until vegetables just start to become tender.
Add 2 quarts broth, 2 tablespoons chopped thyme, remaining 2 teaspoons of salt, and 2 bay leaves. Bring to a boil over medium-high heat, then reduce to a simmer. Cook 20 minutes until vegetables are tender.
Add noodles, 1 cup chopped asparagus, and 1 cup peas and stir to break-up any noodles that might be stuck together. Simmer until noodles are tender, but not mushy, about 4 to 5 minutes more. Taste and add more salt and pepper if necessary.
Serve with a generous sprinkling of dill, scallions, and parmesan cheese on top of each bowl.