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Pour the olive oil into a multi-functional pressure cooker (such as an Instant Pot®) and select the Saute function. Cook onion, carrot, celery, and bay leaf until the vegetables are soft and the onion has turned translucent, about 5 minutes.
Add chicken broth, chicken thighs, chicken breasts, thyme, marjoram, salt, and pepper. Close and lock the lid. Select high pressure according to manufacturer's instructions; set timer for 9 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Carefully remove the chicken pieces with tongs and place them in a bowl to cool slightly; discard bay leaf.
Mash butter with the flour to make a smooth paste; set aside.
Combine flour, baking powder, and salt in a bowl for the dumplings. Cut in cold butter until mixture is the texture of cornmeal. Stir in parsley and set aside.
Shred cooled chicken and return to the pot. Taste the broth and adjust the seasoning if needed. Add peas and green beans. Stir in the flour-butter paste. Select Saute function to bring broth back to a boil.
Pour buttermilk into the dumpling mixture and stir until combined. Drop the dumpling dough by heaping spoonfuls on top of the stew; a small cookie scoop works well.
Cover pot with the lid, leaving the steam vent open. Select Slow Cooker function and simmer on Low, covered, until dumplings are cooked through, 10 to 12 minutes. A skewer inserted in the center of a dumpling should come out clean.