Heat the oven to 400 degrees F and arrange a rack in upper third.
Pat the pork chops dry with paper towels and put on a cutting board. Using a small paring knife, cut a 2-inch opening horizontal to the cutting board, halfway up the meat in each chop. Move the knife back and forth in a fan motion to open the interior of the pocket without significantly widening the opening. (You don't want to pierce through the meat but simply create a pocket for the stuffing.) Brush the exterior of the chops with oil, season well with salt and freshly ground black pepper, to taste, and set aside until the stuffing is ready.
Heat 1 tablespoon of oil in a large ovenproof frying pan over medium-high heat. When the oil shimmers, add the shallots and cook until softened. Add the mushrooms, season well with salt and freshly ground black pepper, and cook, stirring rarely, until browned, about 10 minutes. Stir in the parsley and remove to a plate to cool slightly. Wipe out the frying pan and return it to the stove over medium-high heat. Taste the stuffing and adjust the seasoning, as necessary. Stuff half of the mushroom mixture into the pocket of each chop. If the filling is falling out, secure the chops closed with toothpicks.
Add the remaining 1 tablespoon oil to the pan over medium heat. When the oil shimmers, arrange the stuffed chops in the pan and cook until browned, about 4 minutes. Flip the chops to the other side and transfer the pan to the oven. Bake until the pork is cooked and an instant-read thermometer, inserted into the center of the chops, registers 160 degrees F, about 12 to 15 minutes.
Remove the pan from the oven and set the pork aside on serving plates to rest. Return the pan to the stove over medium heat and add the vermouth. Cook, scraping the bits from the bottom of the pan with a spatula, until the alcohol smell is cooked off and is slightly reduced, about 1 minute. Remove the pan from the heat, add the butter, and let it melt. Taste the sauce and adjust the seasoning. Drizzle the sauce over the chops and serve with a gratin, mashed potatoes, or polenta, if desired.