Preheat oven to 400 degrees F (200 degrees C).

Heat olive oil in a large ovenproof skillet over high heat until lightly smoking. Season filets to taste with salt and pepper. Sear until golden brown on both sides, about 3 minutes per side. Transfer to oven, and cook to desired doneness, about 12 minutes for medium-rare. When done, allow steaks to rest out of the pan for 5 minutes.

Meanwhile, melt 3 tablespoons butter in a saucepan over medium-high heat. Stir in ginger and garlic, and cook until translucent and aromatic, but not browned, about 1 1/2 minutes. Add shiitake mushrooms and cook 3 to 4 minutes, until softened. Pour in the sake and mirin and simmer until reduced by half. Melt the remaining 3/4 cup butter, then reduce heat to medium-low, and cook until the butter browns, 6 to 8 minutes. Once browned, season to taste with salt and pepper, and stir in the chives. Spoon sauce over steaks to serve.