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Set pressure cooker to SAUTE. Drizzle with 1 tablespoon oil. Brown chicken breasts on 1-2 minutes on each side. Transfer to a plate and set aside.
Add 2 tablespoons butter to the pot. Stir in carrots, onions, and garlic for 1 minute til fragrant.
Sprinkle in flour, stir 2 minutes til foamy and white streaks are completely gone. Gradually add chicken broth, italian seasonings, stirring constantly until completely incorporated.
Return chicken to the pot and add salt and pepper. Cover, lock lid into place, and turn vent to SEALED position.
Set to PRESSURE COOK or MANUAL for 5 minutes. (It will take several minutes for the pot to come to pressure and begin counting down from five minutes, meanwhile prepare the dumplings)
Prepare the dumplings by combining flour, baking powder, salt, and cheese in a large bowl and stir til combined. Gradually stir in the heavy cream and butter until a soft dough forms.
When pressure cooking time is complete, quick release (immediately turn the vent valve to VENT and remove the lid once the float valve drops). Remove chicken and shred into chunks with two forks (don't shred too finely) and return to pot. Stir in heavy cream.
Roll 2-3 tablespoons of the dumpling dough loosely in your hands and drop into the pot. Repeat with remaining dough, being sure to drop each dumpling into its own space and not on top of another dumpling.
Cover and lock lid, turn valve to SEALED and set to PRESSURE COOK or MANUAL for 5 minutes (again). Once time is complete, quick release again and remove the lid when the float valve drops.
Garnish with cracked black pepper and fresh parsley if desired and serve.