Grate the potatoes and shallot using your food processor. Add to a colander and squeeze dry using a clean tea towel, making sure to get rid of all of the moisture. Place the potatoes and shallots in a bowl along with the cheese, flour, parsley, egg and some salt and pepper. Mix well.
Heat about 1/4 cup peanut oil in a large cast-iron skillet until hot. Scoop 1/4 cup potato mixture into the skillet and flatten with a spatula. Cook until golden and crisp, 2 to 3 minutes, and then flip and cook another 2 minutes. Remove to a paper-towel-lined baking sheet to drain. Repeat in batches.
Cook's Note: To keep hash brown cakes warm before serving, place in a single layer on a wire rack set over a baking sheet and place in a 200 degree F oven.