Season both sides of the chops with a 1/4 teaspoon of the salt and the pepper to taste; melt half of the butter and the oil in a heavy skillet over medium heat; add the chops and cook until slightly pink inside, about 5 minutes per side.
Transfer the chops to a heated platter; discard the excess fat from the pan and add 1/2 cup of the vinegar and the sherry, stirring and scraping the pan to deglaze; add the chicken broth and heat to a boil over medium heat; cook until the sauce has reduced and thickened slightly, about 10 minutes; remove from the heat and keep warm.
Meanwhile, cook the bacon until crisp in a separate skillet over medium heat; discard all but 2 tablespoons of the bacon fat; add the remaining vinegar, stirring and scraping to deglaze the pan; add the cabbage and increase the heat to high; cook, stirring, until heated through and wilted, about 5 minutes; season with remaining 1/2 teaspoon of the salt and more pepper to taste.
Return the pan of sauce to medium heat; whisk in the mustard and the remaining butter; divide the cabbage/bacon mixture among 4 serving plates and place a pork chop on each bed of cabbage; spoon the sauce around and over each chop.
Garnish with chopped tomatoes and sprinkle with the chives.