In a medium bowl, whisk together the eggs, mayonnaise, Dijon mustard, lime juice, Old Bay seasoning, and salt until thoroughly combined. Stir in green onions and cilantro. Then add in panko bread crumbs and cornmeal. Stir until well blended. Fold in corn kernels and crab meat until evenly mixed.
Using a 1/4 cup measuring cup as a guide, divide the crab mixture into 12 equal portions; shape each portion into a 1/2-inch thick patty.
Place a large non-stick fry pan over medium-high heat and coat with a thin film of oil. Working in batches, gently place about 4 crab cakes at a time in the pan. Fry until golden on one side, 3 to 4 minutes. Using a spatula, carefully flip the cakes over and fry until browned on the other side, 3 to 4 minutes more. Transfer the cakes to paper towels to drain briefly. Repeat with the remaining cakes, adding more oil to the pan as needed.