Place racks in the center and upper third of your oven. Preheat the oven to 400 degrees F. Generously coat a rimmed baking sheet with nonstick spray or line with parchment paper.
Place the cauliflower rice in a microwave-safe bowl. Microwave on high, uncovered, for 2 minutes.
Spread the cauliflower onto a clean, dry kitchen towel and let cool for 5 minutes; keep the cauliflower fairly centered on the towel.
Bring the sides of the towel up and around the cauliflower to make a purse. Squeeze as much moisture out of the cauliflower as possible. The drier it is, the crispier the cauliflower hash browns will be.
To a large, dry mixing bowl, add the egg. Lightly beat it with a fork, then stir in the cheddar, Parmesan, salt, black pepper, garlic powder, onion powder, and smoked paprika.
Add the cauliflower to the bowl and mix to thoroughly combine.
Divide the cauliflower mixture into 8 equal portions. Working one portion at a time, squeeze the portion firmly into a ball with your hands, then gently flatten with your hands to shape into patties that are about 1/2-inch thick. Arrange the patties evenly on the baking sheet, leaving at least 1 inch of space between each.
Bake on the center rack until lightly browned and set, about 15 minutes, rotating the pan 180 degrees halfway through.
Turn the broiler to low. Transfer the pan to the upper-third rack and bake until crispy on top, about 5 minutes. Watch very carefully so that the hash browns do not burn.
Let cool on the baking sheet for 5 minutes to set. Enjoy hot.
TO STORE: Refrigerate hash browns in an airtight storage container for up to 4 days.
TO REHEAT: Rewarm leftovers on a baking sheet in the oven at 350 degrees F.
TO FREEZE: Bake as directed, then let cool to room temperature. Lay hash browns in a single layer on a parchment-lined baking sheet, and freeze until solid. Transfer the frozen hash browns to an airtight freezer-safe storage container, and freeze for up to 3 months. Reheat from frozen.