Heat oil in heavy medium saucepan over medium-low heat. Add shallots. Cover and cook until shallots are deep golden brown, stirring occasionally, about 20 minutes. Cool.

Whisk together mayonnaise and sour cream in medium bowl to blend. Add blue cheese. Using rubber spatula, mash mixture until smooth. Stir in caramelized shallots. Season dip to taste with salt and pepper. Cover dip and refrigerate until flavors blend, about 2 hours. (Can be made 2 days ahead. Keep refrigerated.) Serve dip chilled or at room temperature.