Heat olive oil in a skillet over medium heat; cook and stir garlic until fragrant, about 1 minute. Add Cajun seasoning; cook and stir until garlic is evenly coated, about 1 minute. Mix butter into garlic mixture and cook until bubbling; reduce heat and add sugar, cayenne pepper, and hot sauce. Mix well and remove skillet from heat to cool, about 10 minutes.
Mix yogurt, cucumber, Parmesan cheese, salt, and lime juice together in a bowl; add garlic mixture and stir until fully blended.