In a large bowl, whisk together the eggs and ½ teaspoon each salt and pepper.
Using a food processor with the large grater attachment, coarsely grate the carrots. Add them to the bowl with the eggs and toss to coat. Fold in the panko, then 2 scallions, the chile, and cilantro.
In a small bowl, combine the lime juice, 1 tablespoon oil, and remaining scallion.
Heat a large cast-iron skillet over medium heat, then add 1 tablespoon oil. Drop 6 spoons of the carrot mixture into the skillet and cook until golden brown and crisp, about 3 minutes per side; transfer to a wire rack. Repeat with the remaining oil and carrot mixture, adding more oil to the skillet, if necessary.
Gently stir the feta into the lime-scallion mixture. Serve over the carrot fritters and serve with green salad, if desired.
