Chicken and Sauce:
1 ½ pounds chicken (about three medium-size chicken breasts)
2 cups low-sodium chicken broth
1 tablespoons olive oil
1 onion , chopped fine (about 1 cup)
3-4 carrots, peeled and sliced (about 1 cup)
2-3 stalks celery, chopped (about ½ cup)
½ teaspoon salt
½ teaspoon pepper
4 tablespoons (1/2 stick) butter
½ cup flour
1 cup milk
3/4 cup frozen peas
2 cups all-purpose flour
2 teaspoons baking powder
3/4 teaspoon salt
½ teaspoon pepper
1/8 teaspoon cayenne pepper
6 tablespoons butter, cut into 1/2-inch cubes and chilled
3/4 cup freshly grated Parmesan cheese
1 cup heavy cream
To prepare the chicken, place the chicken and broth in a large Dutch oven or pot and bring it to a simmer over medium heat. Cook the chicken until it is just done (don’t overcook or it will dry out) – about 10-12 minutes. Alternately, if you are planning in advance, you can place the chicken and broth in a slow cooker and cook on high for 3 hours or low for 5 hours. Transfer the cooked chicken to a medium bowl. Pour the broth through a fine-mesh strainer into a liquid measuring cup and reserve. Return the pot to the stovetop (no need to wash if you used it to cook the chicken).
Preheat the oven to 400 degrees. In a large bowl, combine the flour, baking powder, salt, pepper, and cayenne. Sprinkle the chilled butter pieces over the top of the flour. Using your fingers or a pastry cutter, rub the butter into the flour mixture until it resembles coarse crumbs. Stir in the Parmesan cheese. Add the cream and stir until just combined. Crumble the mixture into irregularly shaped pieces onto a baking sheet lined with parchment paper. Bake the crumble topping until fragrant and starting to brown, about 15 minutes, tossing halfway through. Set aside.
For the filling, heat the olive oil in the now-empty pot over medium heat. Add onion, carrots, celery, salt and pepper. Cover and cook, stirring occasionally, until just tender, about 5 minutes. While the vegetables are cooking, shred the chicken into small bite-size pieces. Transfer the cooked vegetables to the bowl with the chicken; set aside.
Over medium heat melt the butter in the empty pot. Stir in the flour and cook for one minute, stirring constantly. Slowly whisk in the reserved chicken broth and milk. Bring to a simmer and stir occasionally until the sauce thickens, about 5 minutes, taking care not to burn the bottom. Add 1/2 teaspoon salt and 1/2 teaspoon pepper. Remove from the heat and stir the chicken, vegetables and peas into the sauce.
Pour the mixture into a 9×13 pan. Scatter the crumble topping evenly over the filling. Place the 9X13-inch pan on a rimmed baking sheet and bake at 400 degrees until the topping is well browned, about 15-20 minutes. Let the casserole stand for 10 minutes before serving.