Make the Calabrian chile butter. Beat together the softened butter with chopped chilies until well combined. Set aside.
Heat 2 tablespoons oil in a 10-inch skillet over medium heat. Add the leeks, season with salt and pepper, and saute until softened and browned, about 7-8 minutes. Remove from heat and transfer the leeks to a large bowl.
In the same pan, heat ½ cup oil. Add the sliced potatoes and garlic. Season with salt and pepper and continue to cook until the potatoes are tender, carefully turning them over throughout the cooking process. If the potatoes break easily, they are overcooked. Remove from heat and drain the excess oil, reserving the oil.
Transfer the potatoes to the bowl with the leeks. Add the eggs and season with salt and pepper. Let sit for 15 minutes.
Heat 1 tablespoon reserved oil and 2 tablespoons Calabrian chile butter in the same skillet over medium heat. Pour the potato egg mixture into the pan and cook until the edges have set, about 2 minutes. Reduce heat to low and continue to cook for 5 minutes.
Insert a rubber spatula into the pan and run it around the edges to loosen the tortilla. Turn off the heat and place a plate on top of the pan. Carefully invert the pan so that the tortilla is on the plate.
Heat the remaining 2 tablespoons Calabrian chile butter in the pan. Gently, slide the tortilla back into the pan so that the bottom is now facing up. Continue to cook the Spanish tortilla for 4-5 minutes or until almost set. The tortilla should still be slightly runny in the center.
Remove from heat and let cool slightly. Meanwhile, heat 2 tablespoons oil in another skillet over medium-high heat. Saute the oyster mushrooms until browned and caramelized. Season with salt and pepper and keep warm.
Toss the arugula with juice of ½ lemon, olive oil, manchego, dill, parsley, salt, and pepper. Top the warm Spanish tortilla with the sauteed mushrooms and arugula salad. Serve immediately.
Notes
The Calabrian butter can be prepared up to 1 week in advance.