In a medium saucepan, heat olive oil and butter over medium heat and add onion, shallots, scallion whites, garlic, salt, and pepper. Cover and cook, stirring occasionally, for 20 minutes or until onions are completely soft. (Do not let the onions caramelize; if they begin to brown, add a little water to the pan.) Remove one-third of the onions from the pan and set aside. Place remaining onions in bowl of a blender and puree until smooth.

Combine reserved onions with pureed onions in a small bowl. Add mayonnaise, creme fraiche, lemon juice, and scallion greens and mix well. Refrigerate until chilled, at least 2 hours and up to overnight. Serve with Parmesan crackers.