Preheat an oven to 350 degrees F (175 degrees C). Lightly grease a baking dish.
Fill a saucepan with lightly salted water, and bring to a rolling boil over high heat. Once the water is boiling, add the green beans, and return to a boil. Cook uncovered until the green beans are crisp-tender, 3 to 4 minutes. Drain the beans, reserving 2/3 cup of the cooking liquid. Spread the beans into the greased baking dish.
Heat the olive oil in a large skillet over medium-high heat. Add the pork chops, and cook until golden brown on each side, about 3 minutes per side. Season the browned pork chops with salt, pepper, sage, and garlic, then arrange over the green beans.
Pour the reserved bean-cooking liquid in the skillet and stir the browned bits until dissolved. Pour over the pork chops. Place the tomato slices over the pork chops, and cover each with a slice of Cheddar cheese.
Bake in the preheated oven until the cheese melts and is golden brown, 20 to 25 minutes. An instant-read thermometer inserted into the center of the pork chops should read 145 degrees F (63 degrees C).