Preheat oven to 425°F (218°C) and place rack on the top rung. Spray a 10.25-inch cast iron skillet with cooking spray.
Dry shredded potatoes thoroughly with paper towels and place in a large bowl. Toss with melted butter, salt, and pepper.
Heat cast iron skillet on medium heat on the stovetop. Press potatoes evenly into the skillet to form a ½-inch thick crust using a flat-bottomed measuring cup. Cook for about 8 minutes, pressing down as needed, until the bottom starts to brown.
Spray the top of the crust with cooking spray and bake for 15 minutes or until lightly golden and crispy.
While the crust bakes, heat another large skillet on medium high heat. Melt 1 tablespoon of butter and sauté broccoli until softened, 1-3 minutes, stirring constantly. Remove to a plate. Add remaining 1 tablespoon of butter, ham, and mushrooms, and sauté for 1 minute, stirring constantly. Remove to the plate.
In a bowl, whisk together eggs, coconut cream or heavy cream, chives, salt, pepper, and nutritional yeast (if using).
Assemble: Lower oven temperature to 350°F (177°C). Spoon broccoli, ham, and mushrooms evenly over the crust. Pour in the egg mixture and spread evenly.
Place the quiche in the oven and bake for about 30 minutes, or until the top is lightly golden and the center is set. To ensure it's fully cooked, check the middle with a knife and an instant read thermometer—the internal temperature should reach at least 165°F (75°C) for safe eating. Once done, remove from the oven, let it cool slightly, and cut into 8 slices.
Remove quiche leftovers from the cast iron skillet and store in an airtight container for up to 4 days or freeze for up to 4 months. View “Storage and Reheating Instructions” section above for more detailed instructions.
