To a very large pot, add the beef, 2 tablespoons of the salt, and 4 quarts of water. Bring to a boil, then turn the heat to medium to maintain a strong simmer. Cook, skimming any scum that rises to the surface, until the broth has reduced to half its initial volume and the beef flakes easily, about 3 hours. Using a spider skimmer or medium strainer, transfer the beef to a bowl. When cool enough to handle, flake into bite-size pieces, discarding the bones.
While the beef cools, ladle 1 cup of the broth into a small bowl and set aside. To the pot, add the carrots, potatoes, 2 teaspoons of the salt, and (if the vegetables are not submerged) enough water to cover by ½ inch. Bring to a boil, then turn the heat to medium and simmer until the vegetables are soft, about 25 minutes. Add the reserved beef and return to a boil. Using a skimmer, transfer the beef and vegetables back to the bowl and tent with foil. Turn off the heat.
In a small serving bowl, use a fork to mash one of the potato quarters. Stir in the garlic, remaining ½ teaspoon salt, and 1½ cups of the hot broth. Place in the center of a very large rimmed serving platter.
In a large bowl, using your hands or a wooden spoon, mix the flour and reserved broth (from step 2) until a shaggy mass forms. Transfer to a work surface and knead, adding flour if necessary to prevent sticking, until the dough is smooth and springs back when pressed with a finger, about 10 minutes. Cover loosely with plastic wrap and set aside for 15 minutes.
Using a bench scraper or knife, quarter the dough, then roll each piece into a ¼-inch-thick rectangle (the dimensions aren’t important at this stage). Cut each rectangle lengthwise into 2-inch-wide strips, then cut each strip crosswise on the bias into ½-inch-by-2-inch parallelograms. (Repeat with any dough scraps until all the dough is used.) Using your index, middle, and ring fingers, press hard into the far edge of one pasta piece; then, pressing continuously into the work surface, drag the pasta toward you to make an irregular, slightly open scroll. Transfer to a lightly floured baking sheet. Repeat with the remaining pieces, occasionally tossing and dusting with flour if you notice the pieces begin sticking together.
Bring the broth back to a boil. Add the pasta and boil, stirring frequently, until the pieces are puffed and soft (but not mushy), 8–9 minutes. Meanwhile, arrange the beef and vegetables on one half of the platter around the garlic sauce.
Using a spider skimmer or medium strainer, transfer the pasta to the other side of the platter, then moisten the meat, vegetables, and pasta with 1 cup of the broth. Serve immediately, passing additional broth to moisten the meat if desired.