Season the chicken on both sides with salt and pepper. Place it in a medium saucepan with the chicken broth. Bring to a very gentle bubble with the lid slightly cracked and cook for 15 minutes, until the chicken is cooked through. Do not let it boil rapidly. Remove the chicken, shred it with forks, and reserve the broth.
Meanwhile, melt the butter in a wide 12-inch pan over medium heat. Add the onions, celery, and carrots and cook for 5 to 6 minutes. Stir in the garlic and seasonings, then cook for 1 more minute.
Add the flour and stir to coat the vegetables. Cook for 2 minutes, until the raw flour smell is gone.
Pour in about two-thirds of the reserved chicken broth in small splashes, stirring well after each addition. Add the half and half the same way.
Stir in the chicken bouillon and soy sauce. Add the shredded chicken and mix until combined.
If the liquid was added gradually, the gravy should already be fairly thick. If you want it thicker, bring it to a gentle boil, then reduce to a simmer.
If desired, stir in some or all of the remaining broth. Keep in mind that the biscuits will absorb some moisture as the dish bakes.
Add the frozen peas and stir until combined and heated through. Remove from the heat.
If your skillet is not oven safe, transfer the filling to a lightly greased 8 x 8-inch baking dish.
Arrange the biscuits on top and bake according to the biscuit instructions you are using, until the biscuits are golden and set in the center. For a crisper bottom, bake the biscuits separately on a baking sheet and place them over the filling after baking.
If you want more browning, brush the biscuit tops with butter and bake at 450°F for 3 to 5 more minutes.