There are a variety of synonyms for the phrase "gave a big cigar" that you can use depending upon your context and intention. Some possible alternatives include "presented a large cigar," "offered a hefty cigar," "bestowed a massive stogie," "granted a huge smoke," "donated an enormous cigar," "conferred a grandiose cigar," "yielded a substantial cigar," or "bequeathed a humongous cigar." All these phrases communicate the idea of giving someone a large, impressive cigar, indicating either respect, gratitude, or goodwill towards the recipient. They can work well in different contexts, such as in a celebratory or congratulatory event, or when showing appreciation or affection towards someone who enjoys smoking cigars.