Preheat the oven to 400°F (200°C). In a medium mixing bowl, combine the parsley, scallions, jalapeño, cornbread, egg, 1/2 teaspoon salt, and a few grinds of black pepper. Slice a deep pocket, about 2 inches wide, into the side of each pork chop. Gently fill each pocket with the prepared cornbread mixture.

Warm the vegetable oil in a large skillet over medium-high heat. Lightly season the exterior of the pork chops with salt and pepper. Sear the chops in the skillet, cooking in batches to avoid overcrowding, until golden brown on both sides, about 2 minutes per side. Transfer the seared pork chops to a rimmed baking sheet, reserving the skillet, and bake them in the preheated oven until fully cooked, approximately 8 to 10 minutes.

While the pork chops bake, bring a medium pot of salted water to a boil. Add the green beans and blanch them for about 3 minutes, until they achieve a crisp-tender texture. Drain the beans and return them to the pot. Add 2 tablespoons of butter, seasoning with salt and pepper to taste. Toss the beans to coat evenly.

Remove the pork chops from the oven and pour any pan juices from the baking sheet into the reserved skillet. Place the skillet over medium heat and sprinkle the flour over the juices. Cook for 1 minute, whisking continuously to incorporate. Gradually add the chicken broth, deglazing the pan and stirring to dissolve any browned bits. Continue whisking until the sauce thickens, about 2 to 3 minutes. Take the skillet off the heat and whisk in the remaining 1 tablespoon of butter, along with 3/4 teaspoon of black pepper. Drizzle the sauce over the pork chops and serve alongside the green beans.