Melt the butter in a large heavy-bottomed pot over medium-high heat. Add the onion, celery, carrots, and garlic. Cook them, stirring often, until everything softens and smells like you’re doing something right. This takes about 6 to 8 minutes.
Add the shredded chicken, chicken broth, evaporated milk, and the can of cream of chicken soup. Stir it all together and bring it to a gentle simmer. Season with salt and black pepper. Taste it. Adjust it. Taste again.
Cut each biscuit into four pieces. Toss the biscuit pieces lightly in flour. This helps them turn into actual dumplings instead of soggy bread.
Add the floured biscuits to the pot. Gently press them under the broth so they get cozy. Put the lid on and let them cook for about 8 to 10 minutes. I usually lift the lid once or twice to break them apart if they clump together and press them back down.
Check one dumpling to make sure it’s cooked through and not doughy in the middle. Once they’re done, serve immediately. This is not a dish that likes waiting.