Make the creamed chicken:
Place chicken, carrot, celery ribs, chicken stock or broth, onion, parsley, thyme, and bay leaf in a Dutch oven. Add water to cover the pieces by 2 inches. Bring to a simmer, then reduce the heat so the liquid barely bubbles.
Partially cover and cook until the meat releases clear juices when pierced with a fork, 20 to 25 minutes for bone-in chicken and about 10 minutes for boneless chicken. Breast meat should reach an internal temperature of 165F and thighs and drumsticks should reach 175 to 180F.
Remove the meat and let it cool. Remove any skin or bones and either cut or shred the chicken into bite-sized pieces.
Skim the fat from the poaching liquid and strain the broth. Set aside.
Creamed Chicken continued:
Melt butter in a large saucepan over medium-low heat. Add flour and whisk until smooth. Cook, whisking constantly, for one minute. Take off the heat. Whisk in two cups of the reserved poaching liquid and whisk until smooth. Whisk in milk or half and half.
Place the pan back on the heat, increase the heat, and bring the mixture to a simmer, whisking constantly. Scrape the inside of the pan and whisk vigorously to break up any lumps. Let simmer for one minute. Stir in cooked chicken, bring to a simmer, and cook for one minute more. Take off the heat and season to taste with lemon juice, salt, and pepper.
Make the buttermilk biscuit dough:
In a large mixing bowl, add flour, baking powder, salt, and baking soda and whisk thoroughly to combine. Cut in the butter using a pastry blender or your fingers to work it into the flour until it is in small, flattened pieces and the mixture is crumbly.
Stir in buttermilk just until the dough comes together.
Transfer the dough to a lightly floured surface and knead very briefly just to bring the dough together. Pat the dough out to a 3/4 inch thick round and use a floured biscuit cutter to cut out biscuits as close together as possible. Do not twist the cutter, just push it down into the dough. Set aside until needed (refrigerate if needed, depending on the timing of the rest of the recipe.)
Position a rack in the upper third of the oven. Preheat the oven to 400F. Grease a 13x9in baking dish.
Melt butter in a large skillet over medium-high heat. Add onions, carrots, and celery, stirring often for about five minutes. Stir the vegetables into the skillet with the creamed chicken and add peas and parsley.
Pour the mixture into the prepared dish. Arrange the biscuit dough pieces on top of the mixture, overlapping the biscuits if necessary. Brush top with egg wash.
Bake until the sauce is bubbling and the topping is browned approximately 30 to 40 minutes. Let cool for 15 minutes before serving.