Preheat oven to 450ºF with a baking stone (or baking sheet) placed on the lower rack. Slice sausage rounds into strips, and then crosswise into 1/2" pieces (you can stack and cut a few at a time).
Heat 1 tbsp oil in a 10” oven-proof cast iron skillet over medium-high heat. Add sausage and cook, stirring occasionally, until nicely browned on both sides, 2-3 minutes. Transfer to a plate.
Add another tbsp of oil to skillet and reduce heat to medium-low; add onions and cook until soft and deeply golden, about 25 minutes.
Meanwhile, prepare crust and filling. Place grated potatoes in the centre of a clean kitchen towel, and gather the towel around it to forcefully wring out the excess moisture. Open towel and sprinkle potatoes with 1/4 tsp salt; toss to season evenly.
In a medium bowl, whisk together eggs, milk, cream, 1/4 tsp salt, pepper and nutmeg.
Transfer caramelized onions to plate with sausage, and carefully wipe skillet clean with paper towels. Add 2 tbsp oil and increase heat to medium-high. When shimmering hot, add grated potatoes and press firmly into the skillet so that the entire bottom is evenly covered. Cook until golden brown on underside, about 4 minutes (use a spatula to gently lift it to check). Carefully flip the hashbrown onto a plate (I put a plate upside-down on top of the hashbrown in the skillet and quickly flip the whole thing over, holding the plate firmly to the skillet with oven mitts). Add another 2 tbsp oil to the pan, and gently slide the hashbrown back in to brown the other side, another 4 minutes or so. (It doesn't tear easily, but if it does, don't fret - just piece it back together and press into the pan - no one will ever know.)
Remove skillet from heat and sprinkle hashbrown with the cheese, then the sausage and caramelized onion. Pour the egg mixture over top, and sprinkle with green onion. Reduce oven temperature to 350ºF and bake on baking stone or sheet for about 20 minutes, until filling is set. Serve warm or room temperature.