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. 

While the steaks are finishing in the oven, add the shallots and garlic to the drippings remaining in the skillet, reduce the heat to medium, and cook until fragrant, about 1 minute. Add the mushrooms and season with salt and pepper. Cook, stirring occasionally, until the mushrooms have softened and are lightly golden, about 2 minutes. Deglace the pan with the cognac, taking care to avoid flames if the cognac ignites. Once the flames die down, add the mustard and cream and stir until smooth and thickened, about 1 minute. Stir in the demiglace and Worcestershire; taste and adjust the seasoning if necessary. 

Remove the filets from the oven and allow to rest briefly before serving.

Serve the steaks on top of creamy mashed potatoes, with a side of wilted spinach. Drizzle the sauce over the steak and potatoes.