Knead the dough: Press down with the heel of your hand, then fold in half, turn 90 degrees, and press again. Continue kneading until dough feels soft and elastic, 8 to 10 minutes. If it still feels sticky, incorporate new flour as you go (do not use the flour from before). Wrap dough in plastic wrap and set aside for 30 minutes to 3 hours.
Divide dough into 4 equal pieces. Take out one piece, and rewrap the others. Using the least amount of flour possible, roll out the dough, stretching and turning it as you go and pressing down hard, until it is roughly 9 x 10 and thin enough to read through. Set aside. Roll out a second piece of dough to the same dimensions.
In a small bowl, whisk remaining egg. Tear basil leaves. Brush one side of each piece of rolled-out dough with egg wash. Sprinkle one piece with 1/4 cup cheese and about 1/3 of the basil. Lay the other piece of rolled-out pasta on top, egg side down, stretching to fit at the edges as necessary. Press together by hand, then roll dough firmly until well sealed. With a knife, cut layered pasta into 1-inch-wide strips. By hand, tear each strip into rough 1-inch-long pieces. Repeat with remaining 2 pieces of dough.
In a large saucepan, bring strained broth to a boil over medium-high heat. Add carrot. When carrot is almost done, add zucchini, peas (if using), and pasta all at once. Raise heat, and boil 3 to 4 minutes. Serve right away with remaining cheese sprinkled over the top, and garnished with basil, if desired.