Heat the olive oil in a 10 inch cast iron pan over medium heat. Once hot, add the scallions and mushrooms, cooking until just softened, about 3-4 minutes.
Add the salt, pepper, and smoked paprika, and continue cooking for another minute. Add the kale, and cook, stirring frequently, until kale is completely cooked down.
In a bowl, whisk together the eggs, creme fraiche or sour cream, and ¾ cup of parmesan cheese. Pour egg mixture into the pan, tilting pan slightly so ensure that the egg mixture forms an even layer over the entire pan. Continue cooking over medium heat on the stove until the egg along the edges of the pan has just set, about 2-3 minutes.
Sprinkle the remaining ¼ cup parmesan cheese evenly over the top, then place the pan in the oven and bake until frittata is puffed and just slightly golden on top, about 10-12 minutes.
Remove from oven, sprinkle parsley over the top, cool slightly, then slice and serve.