Combine milk and eggs, beating well. Gradually stir in flour until mixture forms a soft dough. Divide the dough into three equal portions. Roll each portion to 1/16-inch thickness on a lightly floured surface; cut into 1/2- x 4-inch strips. Let stand on floured surface at room temperature 2 hours or until dry.
Combine chicken, rice, onion, salt, and pepper in a large Dutch oven. Add water to cover and bring to a boil. Cover; reduce heat, and simmer 1 hour or until chicken is tender. Remove onion from broth, and discard. Remove chicken from broth; bone, and chop chicken. Return chicken to broth, and bring to a boil.
Drop dried noodles into boiling broth; cook, uncovered, 30 minutes or until noodles are tender.