In a large bowl, lightly beat your eggs. Add in the butternut squash shreds, flour, seasonings and salt and pepper to taste and the cheese. Stir until just combined.
Heat 1 tablespoon olive oil in a large skillet on medium high, and working in two or three batches, spoon about 1/4 cup of the mixture onto the pan. Flatten them out with your spatula and then use it to create a nice round shaped fritter. Cook about 3 minutes per side on medium heat until golden. Transfer to a paper towel lined platter and repeat with the remaining batter.