In a large skillet, melt the butter. Add the onion and cook over moderate heat, stirring occasionally, until deep golden, about 40 minutes. Add the water and scrape up any browned bits. Cook until the liquid evaporates, about 5 minutes. Season the caramelized onion with salt and pepper; keep warm.
Season Rocket Patty generously with salt and pepper. In a large cast-iron skillet, heat the oil. Cook over moderately high heat until deep brown outside and medium-rare within, about 6 minutes per side. During the last 2 minutes, top the burgers with the cheese and cover loosely with foil so the cheese melts. Transfer the burgers to the buns, top with the caramelized onion and serve with a side of lettuce, tomato and pickles.