Cut a piece of parchment paper so that it covers the bottom of the pan you intend to use.

Melt 2 tablespoons butter in a large saucepan over medium heat; add chicken, in batches if necessary, and brown well on both sides, about 5 minutes per side. Remove chicken to a platter and set aside.

Add remaining 2 tablespoons of butter to pan, reduce heat to low, add carrots, celery, and onions to pan, cover with parchment paper (this will help keep the moisture with the vegetables instead of condensing on the pot lid), put a tight lid on the pot, and cook, stirring occasionally, until vegetables are very soft, 18 to 20 minutes. Do not brown; discard parchment paper.

Combine chicken stock, water, and chicken base and add to pot; return chicken to pot along with any accumulated juices. Bring to a boil, adjust heat to a steady simmer, cover, and cook until chicken is very tender, 30 to 40 minutes. Remove chicken and, when cool enough to handle, remove and discard skin and bones; scoop out and set aside a cup of broth for making the dumplings. Shred chicken and return to pot; stir in green peas. Bring to a full boil and begin dropping dumplings into the pot as quickly as possible while maintaining a full boil.

Reduce heat to low and cook at a low simmer for 15 minutes. Taste and add salt and pepper as desired; remove from heat, cover, and allow to thicken for 10 to 15 minutes. Garnish with celery leaves or parsley and sliced green onions.

Dumplings

Combine flour, pepper, and baking soda in a mixing bowl. Stir in melted butter and enough chicken broth to form a smooth dough (you may not need all the broth; return any left over to the pot).

Generously flour a work surface; turn dough out onto surface and pat into a circle. Sprinkle with flour and roll dough into a circle about ⅛-inch thick, turning often to prevent sticking. Cut into 1-inch wide strips, then crosswise into 1½-inch pieces.