In a large bowl, mix the cooked quinoa with the egg, grated potato, pepper, and salt. Heat a large skillet over medium heat and melt 1 1/2 teaspoons of butter in the pan. Scoop 1/3-½ cup of the quinoa mixture into the pan and flatten out the hash brown patty with the back of a spatula. Cook the hash browns for 5-7 minutes on each side until golden brown. Repeat process with the remaining mixture.