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Bring the chicken broth and vegetable broth to a boil in a covered 4 1/2-quart Dutch oven or soup pot over high heat
This should take 8 to 10 minutes
When the broth boils, add the carrots, noodles, and the chicken and its juices
Cover the pot and bring the broth back to a rolling bo
Uncover the pot and stir, breaking up any chicken chunks slightly
Continue to boil, uncovered, over high heat until the noodles are tender, about 6 minutes
Meanwhile, combine the flour and water in a small container that has a lid
Shake well until the lumps disappear
When the noodles are just tender, add the flour mixture and stir constantly until the stew is slightly thick, about 2 minutes
Season with pepper and remove from the heat
Serve in large soup plates or bowls.