Rinse the chicken and place in a large dutch oven or heavy bottomed pot. Add the celery, onion and carrots. Cover with chicken broth and water, cover and bring to a boil. Reduce heat to a simmer and cook for 30 minutes or until chicken is cooked through. Remove the chicken from the pot and let cool for 10-15 minutes.
Meanwhile in a small skillet, heat the olive oil over medium high heat and add the mushrooms. Cook until browned and tender and set aside.
With clean hands, pick the meat off of the chicken, discarding the skin and bones. Transfer the chicken back to the pot and stir in the mushrooms. Place the pot back on the stove and hold it covered at a low simmer.
In a medium bowl, combine the flour, cornmeal, salt, sugar and chives. Whisk to combine. Stir in the milk and half and half to form a dough.
Drop the dumplings by tablespoonfuls onto the simmering liquid, cover (don't peek) and cook for 15 minutes. Use a spoon to open up a space between the dumplings and pour the peas into the opening. Gently stir the peas into the broth and serve.