Sprinkle the steak liberally with salt and pepper. Heat a cast-iron pan over medium-high heat. Add a light coat of oil and place the steak in the pan. Cook the steak to desired doneness, depending on thickness, 3 to 5 minutes per side. Remove to a plate, cover tightly with aluminum foil, and allow to sit for 10 minutes. Thinly slice the steak into strips against the grain.
Using a sauté pan, heat 1 1/2 tablespoons of olive oil over medium heat. Add the onion and thyme and sauté for 10 minutes, until the onions are brown and caramelized, stirring occasionally.
Meanwhile, prepare the Mustard Mayo:
Whisk the ingredients together in a small bowl. Serve at room temperature.
To assemble the sandwiches, spread a tablespoon of Mustard Mayo on the bottom half of each bun. Place a layer of steak strips on top of the mayo, sprinkle with salt and pepper, and top with the caramelized onion rings. Place the baby arugula on top of the onion rings, and cover the sandwiches with the top half of the buns.
Recipe adapted from: www.foodnetwork.com