Heat the olive oil in a large skillet over medium heat. Add the onion and garlic; cook 3 minutes. Add the eggplant, season with salt and pepper and cook until the eggplant begins to soften, about 3 more minutes. Add the rosemary, tomato and vinegar and simmer until the vegetables are tender, about 10 minutes. Remove from the heat and stir in the capers and pine nuts; set aside to cool slightly.
Place a large skillet over high heat. Season the lamb chops with salt and pepper. Working in batches, place the lamb chops in the pan, fat-side down, and cook until the fat begins to render, 4 to 5 minutes. Turn the chops onto their sides and cook until crisp and brown, 4 to 5 minutes per side for medium-rare.
Divide the eggplant caponata and lamb chops among plates and garnish with arugula.