In a large mixing bowl, combine the drained canned tuna, chopped green onions, grated carrot, salt, pepper, eggs, and flour. Use a fork to gradually mix everything together until all ingredients are well blended, ensuring no dry flour remains.

With your hands, take small portions of the mixture and gently shape them into patties, about 2-3 inches in diameter. Place the formed patties on a plate and refrigerate them for about 10 minutes to help them firm up.

Heat a non-stick skillet over medium heat and drizzle in 2 tablespoons of cooking oil. Once the oil is shimmering, carefully add the patties in a single layer. Cook the cakes for about 2-3 minutes on each side until golden brown and crispy.

While the tuna cakes are cooking, grab a small bowl and mix together the mayonnaise, gochujang, lemon juice, and sugar until smooth and creamy. Adjust the gochujang to achieve your desired spice level.

Once the tuna cakes are golden and crispy, use a slotted spatula to transfer them to a paper towel-lined plate to absorb excess oil. Serve them warm, drizzled with the spicy mayo.