Stir together first 6 ingredients. Gently fold dressing into crabmeat, and fold in breadcrumbs. Season with 1/4 teaspoon salt and 1/4 teaspoon pepper; cover and chill 1 hour.
Puree avocado, sour cream, and lime juice with 1/4 cup water in a food processor until smooth. Season with 1/2 teaspoon salt and 1/4 teaspoon pepper; cover and chill.
Heat oven to 350°. Peel and slice plantains lengthwise using a mandolin or vegetable peeler. (This should yield 40 slices.) Place strips on a parchment paper-lined baking sheet, and lightly grease with cooking spray. Bake at 350° for 2 to 3 minutes or until pliable. Remove from oven.
Arrange 4 plantain slices like the spokes of a wheel on a piece of plastic wrap; repeat 9 times. Place 1/2 cup crabmeat mixture in center of each wheel, and cover crabmeat mixture with ends of plantain slices. Pull plastic wrap around each crab cake, and twist. Chill at least 1 hour.
Fry crab cakes in hot oil in a large skillet over medium-high heat 3 to 4 minutes on each side or until evenly browned. Remove and pat dry. Top each crab cake with avocado aïoli, and serve warm.