In a small saucepan over medium heat, cook garlic and oil, stirring occasionally, until garlic is fragrant and softened, 5 to 7 minutes. Let cool slightly.
Transfer garlic and oil to a blender. Add vinegar, granulated sugar, salt, and pepper and blend until very smooth. Pour into a jar and refrigerate until ready to use.
Prepare a grill for medium-high heat; preheat 5 minutes and clean grates. Generously rub steak with Cajun seasoning.
Grill steak, turning occasionally, until an instant-read thermometer inserted into thickest part registers 130° for medium-rare, about 6 minutes per side. Let cool slightly. Thinly slice steak against the grain.
On a large platter, arrange arugula. Top with steak, cucumbers, bell pepper, avocado, tomatoes, blue cheese, and basil. Drizzle with dressing before serving.
