For the tartar sauce: Mix together the mayonnaise, cornichons and brine, parsley, tarragon, scallion and shallot in a small bowl. Cover and refrigerate until the flavors blend, at least 30 minutes (this can be refrigerated for several days).
For the crab cakes: Line a baking sheet with parchment paper. Gently mix the crabmeat, scallions, mayonnaise, Old Bay, parsley, 4 tablespoons of the breadcrumbs and 1/2 teaspoon salt in a medium bowl, taking care not to break up the crab lumps. Season with white pepper and additional salt, if necessary, to taste. Using a rubber spatula, carefully fold in the eggs until the mixture just clings together. If necessary, add more breadcrumbs 1 tablespoon at a time.
Divide the crab mixture into 12 equal portions and shape each into a round cake, about 2 1/2 inches across and 1 inch high. Lay formed crab cakes on the parchment-lined baking sheet. Cover with plastic wrap and chill at least 30 minutes (these can be refrigerated up to 24 hours).
Put the flour on a plate or in a pie tin. Gently place the crab cakes, one by one, into the flour and dust all over with flour to lightly coat. Set the floured cakes on a wire rack.
In a large nonstick skillet, heat the oil over medium-high heat until hot but not smoking. Gently lay half of the floured crab cakes in the skillet and cook until the outsides are crisp and browned, 4 to 5 minutes per side. Repeat with the remaining crab cakes.