Set up the griddle with a high-heat searing zone, a low-temperature zone, and a 350°F zone.
Cut the top from a bulb of garlic, drizzle with oil, and season with a pinch of salt. Wrap in aluminum foil and roast on the 350°F section for about 40 minutes.
Place a cast-iron skillet on the 350°F section of the griddle to preheat.
Bring the steaks to room temperature and pat them dry. Generously season both sides with salt.
Add 2 tablespoons butter and 1 tablespoon oil to the low-temperature section. Add the mushrooms and cook for 8–12 minutes, until softened and translucent.
Add 2 tablespoons butter to the preheated cast-iron skillet. Add the red onion and cook for 3–5 minutes.
Add the beef broth, garlic paste, thyme, heavy cream, and Worcestershire sauce to the onions. Cook for 15–20 minutes, until reduced into a sauce.
Place the steaks on the searing section of the griddle. Sear for 2–3 minutes per side. Cook until the steaks reach at least 128°F internally.
Remove the steaks from the griddle and let them rest for 6–8 minutes.
Squeeze 4–6 roasted garlic cloves into the mayonnaise and stir until combined.
Toast the bread on the griddle for a few minutes. Thinly slice the rested ribeye steaks.
Spread the roasted garlic mayo over the bread. Layer with sliced ribeye and mushrooms, then spoon the onion sauce over the top.
Add the Swiss cheese and melt it with a kitchen torch or under the broiler. Serve immediately.