Preheat the oven to 400°F.
In a medium cast-iron skillet over medium-high heat, melt the butter. When it sizzles, add the olive oil. Season the meat thoroughly on all sides with salt and pepper, and add them to the sizzling skillet. Sear the filets on all sides until golden brown, 3 to 4 minutes total. Transfer the filets to a small ovenproof baking sheet and place in the oven. Cook to the desired degree of doneness, usually about 5 minutes for medium-rare. Note: exact timing will depend on the thickness of your steaks.
Remove the filets from the oven and allow to rest briefly before serving.
Serve the steaks with a dollop of the compound butter and alongside the Robuchon-Inspired Mashed Potatoes.