First, make the chicken: Add the chicken and broth to a large Dutch oven set over medium heat, and bring to a simmer. Cook for 8-12 minutes, just until the chicken is cooked through. Transfer cooked chicken to a large bowl, and pour the broth through a fine-mesh strainer into a large liquid measuring cup and set aside. Don’t bother to wash out the pot.
Meanwhile, make the topping: Preheat heat oven to 450 degrees F (220 C), and line a large baking sheet with parchment paper or a silicone baking mat. In a large bowl, whisk together the flour, baking powder, salt, black pepper, and cayenne pepper. Cut the 6 tablespoons butter into small pieces, and using your fingers, rub those pieces into the flour mixture until it is coarse and crumbly. Stir in the Parmesan, then add the cream and stir just until combined - do not over mix.
Crumble the biscuit mixture into small, craggy pieces, each about 1/2” or 1” in size, and set the pieces on the prepared baking sheet. Bake for 10-12 minutes, until fragrant and very lightly brown. Set aside to cool.
To make the rest of the filling: Heat 1 tablespoon oil in the now-empty Dutch oven over medium heat. When the oil is hot, add onion, carrots, celery, thyme, ¼ teaspoon salt, and ¼ teaspoon black pepper. Cover and cook, stirring occasionally, until the veggies are tender, 5 to 7 minutes.
While vegetables are cooking, shred or cut the chicken into bite-size pieces. When vegetables are cooked, add them to the bowl with the chicken, and set aside.
In the again-empty Dutch oven, heat the remaining oil over medium heat and add the mushrooms. Cover and cook, stirring occasionally, until the mushrooms release their juices, about 5 minutes. Remove cover and stir in soy sauce and tomato paste. Increase heat to medium-high and cook for 2-3 minutes, stirring frequently, until the liquid has evaporated and the mushrooms are well browned. Transfer mushrooms to bowl with chicken and vegetables; set aside.
Make the roux: Melt the butter in the again-empty Dutch oven over medium heat, then whisk in the flour and cook for 1 minute. Slowly whisk in reserved chicken broth, followed by the milk. Bring the mixture to a simmer, scraping the bottom of the pan to loosen any browned bits, then continue to simmer until sauce becomes thick, 3-5 minutes. Season to taste with salt and pepper.
Remove the roux from the heat and stir in lemon juice and 2 tablespoons parsley.
Assemble: Stir the chicken-vegetable mixture and frozen peas into the sauce, then pour the mixture into a large (13”x9”) baking or casserole dish. Scatter the crumble topping evenly over the filling.
Bake for about 15 minutes, until the filling is bubbly and the topping is well-browned. Allow to cool for 5-10 minutes and sprinkle with remaining tablespoon of parsley before serving. (Or, if you’re out of parsley, skip it, or use something else green for fun. In the photos here, I used rosemary, which turned out to be delicious.)
Leftovers will keep for 3-4 days in the fridge.