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, put a tight lid on the pot, and cook, stirring occasionally, until vegetables are very soft, 18 to 20 minutes. Do not brown. 

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; shred chicken and return to pot. Bring to a simmer, and taste and adjust seasoning. Divide noodles among soup bowls and ladle soup over the top, dividing chicken evenly. Add a soft-boiled egg if desired, and garnish with parsley.