Melt 1 tablespoon butter with 1 tablespoon oil in each of 2 heavy large skillets over medium heat. Sprinkle chops with salt and pepper. Add 3 chops to each skillet and sauté until lightly browned, about 3 minutes per side. Cover skillets and cook until thermometer inserted into center of chops registers 145°F, about 9 minutes. Transfer chops to warm platter; tent with foil to keep warm.
Add half of water, shallot, and garlic to each skillet. Cook until shallot is soft, scraping up any browned bits, about 2 minutes. Add any juices from chops. Whisk in cornichons and mustard. Season with salt and pepper. Pour over chops. Sprinkle chops with parsley.
*Cornichons are tiny brine-packed French pickles, available at specialty foods stores and some supermarkets.
