Whisk vinegar, Parmesan, shallot, garlic, oregano and pepper flakes in small bowl. Microwave until cheese is melted (vinegar will look cloudy) and mixture is fragrant, about 30 seconds. Let cool.
Process mayo, sour cream, basil, and vinegar mixture in food processor until smooth. With motor running, slowly add oil until incorporated, about 1 minute. Season with salt and pepper. Serve. (Dressing can be refrigerated in airtight container for 4 days.).