Prepare Basil Mayonnaise and set aside.
In bowl,stir together crab meat,egg,red pepper flakes,green onion,tarragon,mayonnaise,lemon juice and bread crumbs. Shape into 4 patties.
Melt butter in medium skillet. Add crab cakes in one layer and fry 5 minutes per side, or until golden brown and cooked through. Remove from skillet. Arrange 2 cakes on each plate and pass the Basil Mayonnaise separately.
Basil Mayonnaise
In bowl combine mayonnaise, lemon juice, shallot, basil and black pepper. Mix well and transfer to serving bowl.