Stir together hash browns, carrot, onion, flour, and salt in a large bowl. Stir in beaten eggs.
Heat oil in 12-inch nonstick skillet over medium heat.
Drop four generous 1/4 cup pancakes of potato mixture into skillet; flatten slightly. Cook each pancake until golden brown, turning once, 2 to 3 minutes on each side. Cook remaining pancakes, adding additional oil to the skillet if needed. Drain pancakes on a paper towel.