Remove steaks from the fridge and allow them to come to room temperature.
Melt butter with 1 tablespoon oil in a large skillet over medium heat. Add onion, sprinkle with salt, and cook, stirring occasionally, until tender, about 5 minutes. Add honey and continue to cook, stirring occasionally, until deeply caramelized—25 to 30 minutes. Add sage and cook for 10 more minutes. Watch the onions carefully and add a tablespoon or two of water or broth if they get too dry or start to cook unevenly.
While onions are cooking, prepare steak: Preheat the oven to 425ºF. Warm a cast-iron skillet over high heat until hot. Pat steaks dry; season generously with salt and pepper. Add remaining 1 tablespoon oil to the skillet. Add steaks and sear until nicely browned on both sides, 2 to 3 minutes per side.
Transfer to the oven and cook until an instant-read thermometer stuck into the thickest part of a steak reads 130ºF, 5 to 10 minutes (depending on thickness). Remove the skillet from the oven. Transfer steaks to a cutting board; cover and let rest for 5 to 10 minutes.
Whisk arrowroot into broth until incorporated. Reduce heat on stove to medium-low. When onions have caramelized, add broth mixture one tablespoon at a time and cook, stirring, until the mixture has the consistency you want. Sprinkle with cheese and continue to cook, stirring, until the sauce has thickened and the cheese has melted—about 1 minute.
Place each steak on a plate. Top each with half of the onion mixture and serve.