Heat the oven to 400°F. Using a spice grinder or a mortar and pestle, coarsely grind together quinoa, cumin, and salt; set aside.
Using a food processor with the grating attachment, grate the potatoes. (Alternatively, you can use a box grater.) Remove the grated potatoes from the processor and place them in a mixing bowl.
Let the potatoes rest for 5 minutes. Squeeze the liquid from the potatoes, one handful at a time, and place the potatoes in a separate, clean mixing bowl.
Add the ground quinoa mixture to the squeezed potatoes and mix with your hands until well incorporated. Oil a 13-by-9-inch baking dish with olive oil and place a piece of parchment paper on the bottom, cut to fit exactly.
Pour the potato mixture into the oiled pan and pack it down (make sure to keep it a consistent thickness). Brush the top of the mixture with some more olive oil and cover with another piece of parchment paper placed directly on the potatoes.
Place in the preheated oven and bake for 25 minutes. Remove from the oven and allow to cool to room temperature. Place in the refrigerator to cool completely, about 3 hours.
After 3 hours, remove from the refrigerator and slide a thin, sharp knife around the perimeter of the potatoes to loosen them from the pan. Flip the potatoes onto a cutting board. Trim off any crisp edges and cut into 12 equal pieces.
Fill a frying pan with 1/2 inch of neutral oil (such as vegetable, canola, or peanut) and heat over medium heat. When the oil is hot (about 350°F), fry the hash browns until crisp and golden brown, about 3 to 5 minutes per side. Drain the finished hash browns on paper towels, season with salt, garnish with chopped chives, and serve.