Preheat your oven to 375°F (190°C). Generously grease a 9-inch pie dish or a similar-sized baking dish with the butter or olive oil. This prevents sticking and helps achieve a beautiful golden edge.
Heat the olive oil in a large skillet or Dutch oven over medium heat. Add the diced onion and cook, stirring occasionally, until softened and translucent, about 5-7 minutes. Add the minced garlic and cook for 1 more minute until fragrant. Tip from me: Don't rush this step; properly softened onions build a sweet flavor base.
Add all the fresh spinach to the skillet. It will seem like a lot, but trust me, it wilts down dramatically. Cook, stirring frequently, until the spinach is completely wilted and any excess liquid has evaporated, about 3-5 minutes. Transfer the spinach-onion mixture to a plate or bowl and let it cool for 5-10 minutes. This is crucial—adding hot veggies to the eggs can start cooking them prematurely.
In a large mixing bowl, whisk the eggs vigorously until well beaten and slightly frothy. Whisk in the milk, Dijon mustard (if using), salt, pepper, and nutmeg. Stir in 3/4 cup of the shredded cheese and all of the Parmesan. Finally, fold in the cooled spinach mixture until evenly distributed.
Pour the egg mixture into your prepared baking dish. Smooth the top with a spatula and sprinkle the remaining 1/4 cup of shredded cheese evenly over the surface. Believe me, this top layer of cheese creates a gorgeous, flavorful crust.
Bake in the preheated oven for 35-45 minutes, or until the quiche is deeply golden brown on top, the center is set (it should jiggle only slightly when shaken), and a knife inserted near the center comes out clean. Let the quiche rest on a wire rack for at least 10 minutes before slicing and serving. This allows the structure to firm up for perfect slices.