Thoroughly combine flour and salt in a large mixing bowl; make a well in the center and set aside.
In a medium bowl, whisk together eggs and food coloring. Add vinegar and whisk again; pour into well in flour mixture.
Work dough with hands until all ingredients are completely mixed. On a lightly floured surface, work dough until smooth, adding more flour if dough is sticky. Let dough rest for 20 minutes; divide in half.
Roll out one portion of dough into a circle; continue to roll until thin (noodles will plump up when cooked). Cut dough into long strips, then cut crosswise to desired width. Toss with flour to separate noodles. Set noodles aside and repeat with second portion of dough. Cover noodles loosely with paper towels and let dry for one to 2 hours, or overnight.
Place chicken in a 4-quart saucepan and cover with water. Bring to a boil; reduce heat and simmer until tender, about one hour. Reserving broth, remove chicken and let cool; remove meat from bone and set aside.
Measure reserved broth; add canned broth if necessary to equal 2 quarts. Bring broth to a boil.
Drop noodles into broth a handful at a time, stirring constantly. Reduce heat to a simmer; cover and cook for 20 minutes, stirring occasionally to prevent sticking.
Uncover and add reserved chicken. Cook, uncovered, an additional 20 minutes, or until noodles are tender, stirring occasionally.