Pre-heat the oven to 450 degrees.
Place the chicken breasts at the bottom of a large pot and pour the chicken broth over the top. Cover, turn the heat to medium-high and bring to a simmer. Cook for 10-12 minutes or until the chicken is cooked through. Remove the chicken breasts to a cutting board and shred. Place the shredded chicken in a large bowl and set aside. Pour the liquid from the pot into a cup and also set aside.
Biscuit Topping (make while the chicken is cooking): In a large bowl, combine the flour, baking powder, salt, pepper and cayenne and mix. Add the butter cubes and use your hands to work the butter into the flour mixture entirely until the butter is completely combined and the mixture is crumbly. Add the Parmesan and stir. Add the half and half and stir until doughy. Line a baking sheet with aluminum foil or parchment paper. Drop pieces of the dough into shaggy pieces (1/2” – 1” big) on the baking sheet. Bake for 10-13 minutes until light golden and set aside. Leave the oven on at 450.
In the now-empty large pot, heat 2 teaspoons of the oil over medium heat. Add the onions, carrots and celery and add salt and pepper to taste. Reduce the heat to medium-low and cover. Cook for about 5 minutes, removing the lid to stir occasionally, or until the vegetables are just tender. Add the vegetables to the bowl containing the shredded chicken.
Return the pot to the stove over medium-low and add the remaining 2 teaspoons of oil. Add the mushrooms and cover the pot. Cook, stirring occasionally, for about 5 minutes until the mushrooms have produced liquid. Add the soy sauce and tomato paste and stir together. Raise the heat to medium-high and continue stirring for 5 more minutes or until the liquid has evaporated and the mushrooms are browned. Add the mushrooms to the bowl with the chicken and vegetables in it.
Put the pot back on the stove (this pot is getting a real workout!) and melt the butter over medium heat. Add the flour and whish until combined. While continuing to whisk the mixture, slowly add the reserved chicken broth liquid from step two and whisk until combined and no clumps remain. Continue whisking and slowly add milk as well. Reduce the heat and simmer while stirring until the sauce thickens. Remove the pot from heat and add the parsley, thyme and lemon juice. Add the frozen peas and the chicken/vegetable mixture to the sauce and stir until thoroughly combined.
Pour the mixture a 9 x 13 baking dish and spread evenly. Place the biscuit pieces in a single layer evenly across the top of the chicken mixture. Bake for 15 minutes at 450 or until the mixture is hot and the filling is bubbly.