Step 1Make the aioli by processing the egg yolks, garlic, lemon juice, one-half teaspoon salt and Tabasco sauce in a food processor or blender until thoroughly mixed. Add the water and process 15 seconds. With the machine running, slowly drizzle in the olive oil until the mixture comes together and thickens. Transfer it to a bowl and stir in the tarragon. Cover and refrigerate until ready to use, up to 2 days.
Step 2To make the crab cakes, combine the crabmeat, breadcrumbs, green onions and one-half cup of the aioli. The mixture should hold together. If it doesn't, add a little more aioli. Season with salt to taste.
Step 3Form the mixture into 6 (3-inch) cakes. Sprinkle some more breadcrumbs on a plate and dip one side of each crab cake in the crumbs.
Step 4Using a skillet big enough to hold the cakes comfortably (or cook the cakes in batches), heat the oil over medium-high heat until just smoking. Add the crab cakes, crumb side down, and cook until golden brown, about 4 minutes. Turn the cakes, reduce the heat slightly and cook until golden brown and heated through, about 4 minutes. Drain on paper towels. Serve with the remaining tarragon aioli on the side.