Thick gravy-like sauce in chicken and dumplings usually begins with a roux before broth or any other ingredients are added, but it might need to be thickened at the end of the process if it doesn't thicken sufficiently. Flour and cornstarch are most commonly used to thicken soups and stews. You can't just dump dry flour and cornstarch in the pot as the clumps will swell and gelatinize. Instead, the flour or other thickening agent must be thinned in a slurry before adding it to the simmering broth.
Spoon out the dumplings from the broth and set them aside. The constant stirring required can break down the structure of the dumplings if left in the pot. You might also remove large chunks of chicken or other ingredients that might make it difficult to stir the broth. Spider skimmers and slotted spoons work well for scooping out the solid ingredients and leaving the broth in the pot.
Add flour or cornstarch to a mixing bowl and stir in enough cold water to form a thin paste with no lumps. It takes 3 to 4 tablespoons to thicken thin broth in a large pot, but use 1 or 2 tablespoons of thickening agent for a small pot or if you only want to thicken it slightly. You can always add more later if it's not thick enough.
Pour 1 to 2 cups of hot broth into the cold slurry gradually while stirring constantly with a whisk. Stir until the flour or cornstarch is dissolved. It helps to pour a steady stream of broth with one hand while simultaneously stirring with your other hand.
Pour the thickening slurry into the pot of hot broth in a steady stream. Whisk constantly as you did when adding the hot broth to the cold water and thickening agent.
Turn up the heat to medium-high and bring the broth to a boil, stirring occasionally.
Reduce the heat and simmer for about 5 minutes. After this you can make another slurry with flour or cornstarch if the broth is still much too runny. Don't add more thickener if the broth is just a bit runnier than you prefer as these thickening agents continue to thicken the broth as it cools.
Replace the dumplings or other solids and simmer just long enough to make them hot.