Preheat the oven to 400 degrees and prepare a baking sheet.
Peel potatoes and coarsely grate into a colander. Mix in lemon juice and drain for 5 minutes. In a medium bowl, stir together the carrots and egg. Stir in the flour, salt, and pepper.
In a large (at least 10-inch) heavy nonstick skillet, heat 2 tablespoons oil over medium heat. Spoon 1/4 cup of the mixture loosely in the pan so it forms a pancake. Repeat to make 3 more pancakes. Press on each lightly with a flat spatula to make a cake about 1/2 inch wide. Cook until browned on one side, about 3 minutes. Turn the cakes over and brown the other side, about 3 minutes.
Transfer to the baking sheet. Repeat with remaining pancake mixture, adding 1 to 2 tablespoons of oil to the pan with each batch, depending on how much oil is remaining in the pan.
Bake for 8 to 10 minutes to cook the pancakes through. Drain briefly on paper towels. Season with salt and pepper. Serve while hot, with yogurt and/or applesauce.