On a lightly floured surface, roll out the dough into a round, about 17 inches across and ¼-inch thick. Using a rolling pin, transfer the dough to a 10-inch enameled cast iron skillet and unfurl it into the pan. Gently lift up the dough at the edges and use your hands to nudge it into the base of the pan, leaving an overhang all around. Place on a baking sheet, cover loosely with plastic wrap and refrigerate.
Melt 2 tablespoons (28 grams) of butter in a large Dutch Oven or pot set over medium heat. Add the leeks and cook until soft and wilted, about 4-5 minutes. Add the remaining butter to the pot and stir until melted. Sprinkle the flour evenly over the leeks, and stir well to combine. Cook over medium heat, stirring constantly for 1-2 minutes.
Add the milk and cream and bring to a simmer over medium-low heat, stirring frequently until the mixture simmers and thickens (it should coat the back of a spoon). Season with salt, black pepper, and white pepper. Taste and add more salt and black pepper if desired.
Add the greens in handfuls, stirring occasionally until they’ve started to cook down. Once all of the greens have been added, stir the mixture over medium-low heat until the greens are fully wilted and reduced significantly in volume, 10-12 minutes. Remove pot from the heat and allow to cool for 15-20 minutes.
While the greens cool, preheat oven to 400°F/200°C with the oven rack in the lower third of the oven. Remove the crust from the refrigerator and uncover it. Pile the greens into the pastry-lined skillet, note that they will be mounded quite high. Sprinkle the cheese all over the top of the greens. Fold the excess dough over the outer edges of the filling, letting it pleat naturally as you go (it will cover about ¾ of the filling). Brush the surface of the dough with egg wash, then generously grate parmesan over the surface of the filling and crust.
Transfer skillet to preheated oven and bake for 30 minutes. Then reduce the oven temperature to 375°F/190°C, and bake for an additional 40-50 minutes more, until the crust is deeply golden brown and the surface of the filling is visibly browned. If the crust is browning too much on the surface as it cooks, you can tent the surface of the pie with foil for the remainder of baking.
Cool galette for 10 minutes before slicing and serving warm, or cool completely and serve at room temperature.