In a mixing bowl, whisk together the egg, mayonnaise, minced green onions, lemon juice, tarragon, and red pepper flakes.
Gently fold in the crabmeat and crushed crackers, being careful not to break up the crab too much.
Shape the mixture into four equal-sized patties and let them rest for 5 minutes.
Heat butter in a frying pan over medium heat until melted and hot.
Cook the crab cakes for about 4-5 minutes per side until golden brown and crispy.
Remove from the pan and place on a plate lined with paper towels to absorb excess oil.
Serve hot with your favorite sauce, like tartar sauce or a squeeze of fresh lemon.