If you're new to Busan, choose between dwaeji-gukbap and milmyeon by the kind of meal you want rather than a universal 'best restaurant' list.
Dwaeji-gukbap is a hot pork-and-broth rice meal. Busan City's cultural essay notes that pork cuts and broth recipes vary from shop to shop; chive salad (jeongguji in local speech) and salted shrimp may accompany it. Ask about meat/offal choices, seasoning and side dishes when ordering.
Milmyeon is a Busan-style wheat noodle dish. Depending on the restaurant it may come in cold broth or a seasoned mixed version. For a cool noodle meal, milmyeon may suit you; for warm broth, gukbap is the intuitive choice. Personal tastes differ.
Busan City's source explicitly says the exact origin of pork soup is uncertain, so avoid presenting one wartime origin story as settled fact. Consult Visit Busan's food features for cultural context, but confirm each restaurant's current opening hours, prices, ingredients and allergen information directly.