First, cook the chicken. Place the chicken breasts or thighs and broth together in a large Dutch oven over medium heat. 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.
1.5 pounds (about 750 grams) boneless, skinless chicken breasts or thighs, 3 cups low-sodium chicken broth or stock
While chicken simmers, make the topping. Preheat the 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.
2 cups (240 grams) all-purpose flour, 2 teaspoons baking powder, 3/4 teaspoon kosher salt, freshly-ground black pepper, 1/8 teaspoon cayenne pepper
Cut the 6 Tablespoons butter into small pieces. Use your fingers or two forks to incorporate 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. Add a few extra drops of cream if needed to press in any loose crumbs.
6 Tablespoons (85 grams) unsalted butter, 1/2 cup grated Parmesan cheese, 3/4 cup plus 2 Tablespoons heavy cream
Gently press the biscuit mixture into small, craggy pieces, roughly 2-inches each, and set the pieces on the prepared baking sheet. Bake for 10-12 minutes, until fragrant and very lightly brown. Set aside to cool. Leave the oven on.
After removing the chicken and broth, prepare the rest of the filling. Warm 1 Tablespoon of the oil in the now-empty Dutch oven over medium heat. Add the onion, carrots, celery, thyme, ¼ teaspoon salt, and ¼ teaspoon black pepper. Cook, stirring occasionally, until the veggies are tender, 5 to 7 minutes.
2 Tablespoons olive oil, 1 medium yellow onion, 3 whole carrots, 2 celery ribs, 1 Tablespoon fresh thyme leaves, kosher salt and black pepper
Meanwhile, shred or cut the chicken into bite-size pieces. When the vegetables are cooked, add them to the bowl with the chicken.
In the again-empty Dutch oven, still over medium heat, warm the remaining Tablespoon oil and add the mushrooms. Cook, stirring occasionally, until the mushrooms release their juices, about 5 minutes. Stir in the 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 the bowl with chicken and vegetables.
10 ounces (about 300 grams) small mushrooms, 1 teaspoon low-sodium soy sauce, 1 teaspoon tomato paste
Make the roux. Melt the 4 Tablespoons butter in the again-empty Dutch oven over medium heat, then whisk in the flour and cook for 1 minute.
4 Tablespoons (57 grams) unsalted butter, 1/2 cup (60 grams) all-purpose flour
Slowly whisk in reserved chicken broth, followed by the milk. Bring the mixture to a simmer and scrape the bottom of the pan to loosen any browned bits. Keep bubbling for 3-5 minutes, until the sauce thickens enough to coat the back of a spoon.
1 cup milk
Remove the roux from the heat and stir in lemon juice and parsley. Sample and season with more salt and pepper to taste.
juice of 1 small lemon, 1 Tablespoon chopped fresh parsley leaves
Put it all together. Stir all of the chicken-veggie mixture and the frozen peas into the sauce, then pour the mixture into a 9"x13" baking or casserole dish. Scatter the biscuit crumbles evenly over the filling.
1 cup frozen baby peas
Bake for about 15 minutes, until the filling is bubbly and the topping is well-browned. Let cool for 5-10 minutes, then serve big scoops in bowls.