Put the chicken in a large Dutch oven, cover it with water, add 1 stick of butter and the chicken bouillon, then cook over medium heat until the chicken is tender, about 45 minutes for chicken breasts or about 1 hour and 15 minutes for a whole 3-pound chicken.

Remove the cooked chicken from the broth and let it cool enough to handle, keeping the broth in the Dutch oven.

Take off and discard the skin, remove the bones, then shred or chop the chicken into bite-size pieces and set it aside without adding it back to the pot yet.

For the dumplings, whisk the 3 eggs into the melted stick of butter in a small mixing bowl until combined.

Add the milk to the egg and butter mixture, using about 1/2 cup.

Stir in the salt and pepper, then add 3 cups of the flour a little at a time, mixing as you go, until the dough is rather sticky.

Stir in the 2 drops of yellow food coloring, or use a pinch of turmeric, to give the dumplings a warm golden color.

Heap about 2 cups of flour onto a wooden board or clean countertop, place about 1 cup of the sticky dough in the middle, and work in enough flour so the dough can be rolled without sticking.

Roll the dough very thin, then use a pizza cutter to cut it into strips, repeating with the remaining dough in batches.

Bring the broth back to a rapid boil, making sure there is plenty of liquid in the pot, then drop in the first batch of cut dumplings and turn the heat down to low while rolling and cutting the next batch.

Continue adding the dumplings in batches until all the dough is in the broth, keeping the pot at a low simmer and stirring gently.

After the last dumpling goes in, add the shredded or chopped chicken back to the Dutch oven, stir very gently, then remove the pot from the heat, cover it, and let it sit for about 10 minutes before serving.