1 1/2 pounds boneless, skinless chicken breasts and/or thighs
3 cups low-sodium chicken broth
2 tablespoons vegetable oil
1 medium onion , chopped fine (about 1 cup)
3 medium carrots , peeled and cut crosswise into 1/4-inch-thick slices (about 1 cup)
2 small celery ribs , chopped fine (about 1/2 cup)
Table salt and ground black pepper
10 ounces cremini mushrooms , stems trimmed, caps wiped clean and sliced thin
1 teaspoon soy sauce (see note)
1 teaspoon tomato paste (see note)
4 tablespoons (1/2 stick) unsalted butter
1/2 cup unbleached all-purpose flour
1 cup whole milk
2 teaspoons juice from 1 lemon
3 tablespoons minced fresh parsley leaves
3/4 cup frozen baby peas
2 cups (10 ounces) unbleached all-purpose flour
2 teaspoons baking powder
3/4 teaspoon table salt
1/2 teaspoon ground black pepper
1/8 teaspoon cayenne pepper
6 tablespoons unsalted butter , cut into 1/2-inch cubes and chilled
1 ounce Parmesan cheese , finely grated (about 1/2 cup)
3/4 cup plus 2 tablespoons heavy cream (see note)
FOR THE CHICKEN: Bring chicken and broth to simmer in covered Dutch oven over medium heat. Cook until chicken is just done, 8 to 12 minutes. Transfer cooked chicken to large bowl. Pour broth through fine-mesh strainer into liquid measuring cup and reserve. Do not wash Dutch oven. Meanwhile, adjust oven rack to upper-middle position and heat oven to 450 degrees.
2. FOR THE TOPPING: Combine flour, baking powder, salt, black pepper, and cayenne pepper in large bowl. Sprinkle butter pieces over top of flour. Using fingers, rub 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 parchment-lined rimmed baking sheet. Bake until fragrant and starting to brown, 10 to 13 minutes. Set aside.
3. FOR THE FILLING: Heat 1 tablespoon oil in 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.
4. Heat remaining tablespoon oil in 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 foam begins to form on surface of pan, about 5 minutes. Transfer mushrooms to bowl with chicken and vegetables. Set aside.
5. Heat butter in 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 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.
6. 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.