Preheat oven to 400° F. Remove the pie crust from the freezer and allow to thaw while the oven is preheating. Once oven is preheated, place pie crust on a baking sheet and bake for 10 minutes. When done, remove pie shell from the oven and lower the heat to 375° F.
While the pie crust is pre-baking in the oven, heat olive oil over medium heat in a non-stick skillet. Add broccoli and onion plus kosher salt to taste. Cook for about 5-7 minutes, until the broccoli is soft and the onion pieces are translucent. Remove from heat and set aside.
Add tofu, soy milk, nutritional yeast, mustard powder, nutmeg, freshly ground black pepper and additional kosher salt to taste to your food processor. Process on low until smooth, scraping down the sides as needed.
Add the tofu mixture, broccoli and onion mixture and cheddar-style shreds to a mixing bowl. Use a spoon or silicone spatula to mix the ingredients until well-combined.
Pour the combined mixture into the pre-baked pie crust and bake on 375° F for 40-55 minutes (see notes below regarding cooking time) until the quiche is set. Allow to cool for 10-15 minutes on the baking sheet before slicing and serving.