Season filets to taste with salt and pepper. Heat olive oil in a heavy, oven-proof skillet over medium-high heat until it begins to smoke. Sear steaks until golden brown on both sides, about 3 minutes per side. Transfer steaks to preheated oven, and cook for about 10 minutes, or to desired doneness. When the steaks are done, set aside to rest in a warm place while finishing the sauce.
Meanwhile, melt 3 tablespoons of butter in a saucepan over medium heat. Stir in the garlic and ginger, cook gently until fragrant and translucent, about 1 1/2 to 2 minutes. Add the shiitake mushrooms and 1/2 teaspoon salt, and cook until softened, 3 to 4 minutes. Pour in sake and mirin; reduce by half.
Stir in remaining 1/2 cup of butter. Once melted, reduce heat to medium-low and continue to cook until the butter turns a dark golden brown, 6 to 8 minutes. Season to taste with salt and pepper, stir in chives. Serve sauce over steaks.