Sort through crab; remove and discard any bits of shell. In a bowl, combine crab, 3/4 cup panko, and green onions. Gently mix in 3/4 cup tarragon aioli just until mixture holds together.
Press mixture firmly into eight equal patties about 3 inches wide; set slightly apart on waxed paper or foil. Pour remaining panko into a shallow bowl.
Preheat oven to 200°. Pour 1/3 cup oil into a large frying pan over medium-high heat. When hot, set each crab cake in panko. Using a slotted spatula, turn, pressing gently to coat.
Fry cakes in small batches, using more oil as needed, until golden brown on the bottom, 2 to 4 minutes; turn gently and cook until browned on the other side and hot in the middle, 2 to 4 minutes longer. Transfer cakes as cooked to a rimmed baking sheet in a single layer and keep warm in oven. Discard any remaining panko.
Divide cabbage salad among plates. Set crab cakes next to salad, top each with a dollop of tarragon aioli, and serve immediately, with remaining aioli in a bowl.
Nutritional analysis per crab cake.