For the filling:
3 cups low-sodium chicken broth
1½ lbs. boneless, skinless chicken thighs and/or breasts
2 tbsp. vegetable oil, divided
1 onion, chopped
3 large carrots, peeled and sliced ¼-inch thick
2 ribs celery, chopped
Salt and pepper
10 oz. baby bella or button mushrooms, sliced
1 tsp. soy sauce
1 tsp. tomato paste
4 tbsp. unsalted butter
½ cup all-purpose flour
1 cup milk
2 tbsp. minced fresh parsley
2 tsp. minced fresh thyme
Squeeze of fresh lemon juice
¾ cup frozen peas
For the topping:
2 cups all-purpose flour
2 tsp. baking powder
¾ tsp. salt
½ tsp. pepper
¼ tsp. cayenne pepper
6 tbsp. cold unsalted butter, cut into cubes
½ cup grated Parmesan cheese
¾ cup plus 2 tbsp. heavy cream
To make the filling, place the chicken in a large stockpot or Dutch oven and cover with the chicken broth. Bring to a simmer over medium-high heat and let cook, about 8-10 minutes, until the chicken is cooked through (registering 160˚ F on an instant-read thermometer). Remove the cooked chicken pieces to a bowl and set aside. Transfer the broth to a liquid measuring cup and set aside.
Preheat the oven to 450˚ F. To make the biscuit topping, combine the flour, baking powder, salt, pepper, and cayenne in a large bowl. Whisk to blend. Toss the butter with the dry ingredients and work the butter into the dry ingredients until the mixture is coarse and crumbly. Stir in the Parmesan and then the heavy cream, stirring just until a shaggy dough forms. Crumble the dough onto a lined baking sheet into pieces about ½- to ¾-inch. Bake 10-13 minutes, until light golden.
Heat 1 tablespoon of the oil in the Dutch oven over medium heat. Stir in the onion, carrots and celery. Season generously with salt and pepper. Reduce the heat to medium-low, cover and cook, stirring occasionally until the vegetables are just tender, about 5 minutes. While the vegetables are cooking, shred the chicken into bite-sized pieces. When the vegetables are just tender, transfer to the bowl with the shredded chicken. Return the pot to the heat.
Heat the remaining 1 tablespoon of oil in the pan. Add the mushrooms, cover and cook, stirring occasionally, until the mushrooms have released their liquid, about 5 minutes. Stir in the soy sauce and tomato paste. Increase the heat to medium-high, stirring often, until the liquid has evaporated and the mushrooms are somewhat browned, about 5 minutes. Transfer to the bowl with the chicken and the vegetables. Return the pot to the heat.
Melt the butter over medium heat. When it is fully melted, whisk in the flour so that no clumps remain. Slowly whisk in the reserved chicken broth and milk. Bring the mixture to a simmer, stirring frequently, and cook until the sauce thickens. Remove from the heat and stir in the parsley, thyme and lemon juice. Stir in the chicken-vegetable mixture and the frozen peas. Adjust seasoning to taste.
Transfer the filling mixture to a 9 x 13-inch baking dish. Scatter the crumble topping over the filling. Place the dish on a baking sheet and bake until the filling is bubbling and the top is golden brown, at least 13-15 minutes. Let cool slightly before serving.