Preheat the oven to 350 degrees F. Spray a baking sheet with nonstick cooking spray.
Mix together the flour and 1 teaspoon of the House Seasoning in a small bowl.
Sprinkle the chicken with the remaining 1 teaspoon House Seasoning.
Pour the buttermilk into a shallow dish. Dredge the chicken in the buttermilk, followed by the flour.
Heat 2 tablespoons of the oil in a heavy nonstick skillet over medium-high heat.
Add half of the chicken breasts to the hot oil and cook until both sides are browned, about 3 minutes per side, and then transfer to the baking sheet. Repeat with the remaining chicken breasts.
Transfer the baking sheet to the oven and bake until the chicken is cooked through, about 10 minutes.
Whisk together the cornstarch and 1/4 cup chicken broth until dissolved. Set aside.
To make the gravy , add the remaining 1 tablespoon oil to the same skillet and heat over medium heat.
Add the onions and saute until translucent, about 2 minutes.
Add the remaining chicken broth, scrape the pan drippings with a wooden spoon, raise the heat to medium-high and cook until the mixture begins to bubble, about 2 minutes. Stir in the dissolved cornstarch to incorporate. Bring to a simmer and continue to cook until the mixture thickens, 4 to 5 minutes. Stir in the milk and black pepper and continue cooking over medium-high heat until thickened, 5 minutes longer.
Remove the chicken from the oven and top each piece with 4 teaspoons of the gravy.
