To make the sauce, whisk together all the sauce ingredients until fully combined. Cover and refrigerate until ready to use.
To make the caramelized onions, to a medium skillet over low heat, add the olive oil.
Once heated add the onions and let cook low and slow for about 15 minutes, stirring occasionally.
Add the Worcestershire sauce, salt, and pepper and stir to combine. Cook for a further 15-20 minutes until the onions are nicely caramelized.
While you cook the onions remove the steak from the refrigerator. Sprinkle salt and pepper on the steak and toss to coat evenly.
Let rest for about 20 minutes at room temperature.
Once ready to cook, heat a cast iron or heavy-bottomed skillet over medium-high heat and add the oil.
Add the steak strips in a single layer, if possible, and cook for 1-2 minutes uninterrupted. Flip steak and cook for an additional 1-2 minutes or longer to reach your desired steak temperature. If you need to do this in 2 batches to not over crowd, do so.
Remove from the skillet to a bowl and let the steak rest for 5 minutes before assembling sandwiches.
While the steak rests after cooking you can assemble the sandwiches.
Lightly toast your buns in the toaster and spread a generous amount of the sauce on both sides of the buns.
Once your steak is sliced, add half of the meat and half of the onions to the bottom half of each bun.
Add a handful of arugulas on the top bun and place the top bun on top of the steak and onions. Slice in half if desired and serve warm.