In a deep, wide sauté pan over medium heat, warm the oil; add sliced fennel and salt. Sauté 10 minutes, or until fennel begins to caramelize. Add the red cabbage, water, and apple cider vinegar. Raise heat to high to bring to a quick boil. Cover, reduce heat to low, and cook 20 minutes. Uncover and stir in the mirin and ground fennel seeds. Cook over high heat, stirring occasionally until liquid evaporates. Remove from heat.