Combine eggs, egg yolks, water, and salt in a bowl; beat with an electric mixer until combined. Beat in 2 cups flour. Mix in 1 cup flour with a wooden spoon. Add more flour to make it as thick as you can stir. Divide dough into 4 equal balls.
Generously dust a pastry cloth, a rolling pin, and 4 sheets of waxed paper with flour. Roll each ball of dough on the pastry cloth until 1/8 to 1/4 inch thick. Let dough sit for 45 minutes. Flip each piece of dough (it will be hard to flip) and let sit for another 45 minutes.
Stack pieces of dough together and cut into noodles. Let noodles dry for about 3 hours.
Bring chicken broth to a boil in a large pot. Add noodles and cook, uncovered, until soft with no bite and the broth has thickened, 15 to 20 minutes.