Fillet Mignon with Lyonnaise Potato Crust
4 tablespoons unsalted butter
2 tablespoons vegetable oil
2 medium or 1 large onion, peeled, cut in half through the poles, and thinly sliced crosswise
2 6-ounce fillet mignons, at room temperature
Salt and freshly-ground black pepper
1 medium to large russet potato
1/4 cup grated Gruyere cheese
Adjust the oven rack to about 6 inches below the broiler and preheat the oven to 400 degrees F. Start with the caramelized onions: melt 3 tablespoons of the butter in a large non-stick skillet. Add the onions and a little salt and pepper. Stir well, then put the cover on and cook over medium-low heat for about 15 minutes. Take the lid off, raise the heat to medium, and continue to cook the onions, stirring occasionally, until they’re a deep golden brown. You should end up with at least a quarter cup of caramelized onions. Put the onions in a bowl and set aside.
Salt and pepper the steaks. Put 1-1/2 tablespoons f the vegetable oil in an ovenproof skillet (like a medium cast-iron skillet). Heat on high until the oil shimmers and then you just start to see a little wisp of smoke. Gently lay the steaks in the skillet, lower the heat to medium, and cook for 5 minutes. Turn the steaks over and cook for another 5 minutes, then put the skillet in the oven for 5 – 10 minutes, until an instant-read thermometer reads 125 degrees F. Take the skillet from the oven and transfer the steaks to a plate. Turn on the broiler. Let the steaks rest on the plate for 15 minutes, uncovered.
While you’re cooking the steaks, start the potato. Peel it, then slice it using the 2 mm disk of a food processor, a mandoline, or slice the potato very thinly by hand. Pat the slices dry with a towel. Using the skillet you browned the onions in, melt the last tablespoon of butter and add the last half tablespoon of oil. Crank the heat up and layer the potatoes in the skillet, fitting in as many slices as you can. Cook them until they’re brown on the bottom, then turn them and brown the other side. This shouldn’t take more than 10 minutes.
When the steaks have rested, return them to the skillet you cooked them in. Shingle the browned potatoes on top of the steaks. (Don’t worry if you have too many slices, just put on as many as you need.) Gently spread about 2 tablespoons of the caramelized onions on top, completely covering the potatoes. Then sprinkle on all the cheese. Put the skillet in the oven under the broiler and broil until the cheese melts and barely starts to brown, 1 to 2 minutes. Remove the skillet from the oven, and serve the steaks immediately.