Bring chicken and broth to simmer in a covered Dutch oven over medium heat. Cook until chicken is just done, 8 to 12 minutes. Transfer cooked chicken to a large bowl. Pour broth through a fine-mesh strainer into liquid measuring cup and reserve. Do not wash the Dutch oven. Meanwhile, adjust oven rack to upper-middle position and heat oven to 450 degrees.
FOR THE TOPPING:
Combine flour, baking powder, salt, black pepper, and cayenne pepper in a large bowl. Sprinkle butter pieces over top of the flour. Using fingers, rub the butter into flour mixture until it resembles coarse cornmeal. Stir in Parmesan. Add cream and stir until just combined. Crumble mixture into irregularly shaped pieces ranging from 1/2 to ¾ inch each onto a parchment-lined rimmed baking sheet. Bake until fragrant and starting to brown, 10 to 13 minutes. Set aside.
FOR THE FILLING:
Heat 1 tablespoon oil in the now-empty Dutch oven over medium heat until shimmering. Add onion, carrots, celery, ¼ teaspoon salt, and ¼ teaspoon pepper; cover and cook, stirring occasionally, until just tender, 5 to 7 minutes. While vegetables are cooking, shred chicken into small bite-size pieces. Transfer cooked vegetables to bowl with chicken; set aside.
Heat remaining tablespoon oil in the empty Dutch oven over medium heat until shimmering. Add mushrooms; cover and cook, stirring occasionally until mushrooms have released their juices, about 5 minutes. Remove cover and stir in soy sauce and tomato paste. Increase heat to medium-high and cook, stirring frequently, until liquid has evaporated, mushrooms are well browned, and dark fond begins to form on the surface of the pan, about 5 minutes. Transfer mushrooms to bowl with chicken and vegetables. Set aside.
Heat butter in the empty Dutch oven over medium heat. When foaming subsides, stir in flour and cook 1 minute. Slowly whisk in reserved chicken broth and milk. Bring to simmer, scraping pan bottom with a wooden spoon to loosen browned bits, then continue to simmer until sauce fully thickens, about 1 minute. Season to taste with salt and pepper. Remove from heat and stir in lemon juice and 2 tablespoons parsley.
Stir chicken-vegetable mixture and peas into sauce. Pour mixture into 13 by 9-inch baking dish or casserole dish of similar size. Scatter crumble topping evenly over filling. Bake on rimmed baking sheet until filling is bubbling and topping is well browned 12 to 15 minutes. Sprinkle with remaining tablespoon parsley and serve.