Season both sides of the chicken breasts with salt and pepper. Place the seasoned chicken into a medium saucepan with the chicken broth. Bring to a very gentle bubble with the lid cracked and simmer for about 15 minutes, until the chicken is cooked through. Avoid boiling rapidly, as that can make the chicken tough. Once cooked, remove the chicken and shred it using two forks. Reserve the broth used for cooking.
While the chicken is cooking, melt butter in a wide (12-inch) pan over medium heat. Add the diced onion, celery, and carrots, and sauté for 5-6 minutes until softened. Stir in minced garlic, dried rosemary, dried thyme, ground sage, and onion powder. Toss everything to coat evenly and cook for an additional minute to bloom the flavors. I like to use fresh herbs whenever possible for an even brighter flavor, but dried herbs work well here too.
Sprinkle flour over the sautéed vegetables and herbs, tossing to coat. Cook for about 2 minutes, stirring frequently, until the flour is incorporated and no longer smells raw. Gradually add about 2 cups (2/3) of the reserved chicken broth from Step 1 in small splashes, stirring constantly to avoid lumps. Gradually add the half and half in the same manner, continuing to stir. Add the chicken bouillon cube and soy sauce, stirring until dissolved.
Add the shredded chicken from Step 1 to the gravy and mix thoroughly. If you prefer a thicker gravy, bring the mixture to a gentle boil and then reduce to a simmer. If you’d like a looser consistency, stir in more of the remaining chicken broth, as needed. Add the frozen peas and stir to combine, heating through. Remove from heat. I always taste the filling at this point and adjust the seasoning with salt and pepper as needed.
If your skillet isn’t oven-safe, transfer the filling to a lightly greased 8 x 8-inch baking dish. Arrange the prepared (unbaked) biscuits on top of the filling. Bake in a preheated oven according to your biscuit recipe or package instructions, or until the biscuits are golden and set in the middle. For extra color, you can brush the biscuit tops with butter and bake at 450°F for an additional 3-5 minutes.