Season the steaks with salt and pepper.
Heat a skillet with olive oil until very hot.
Sear steaks for 2–3 minutes per side.
Add butter, garlic, and thyme.
Baste steaks with melted butter.
Remove steaks and rest for 5 minutes.
Slice steaks thinly against the grain.
In the same pan, sauté onions until caramelized.
Add balsamic vinegar and cook one minute.
Mix mayonnaise, mustard, and lemon juice in a bowl.
Toast the bread until crisp and golden.
Spread sauce on bread.
Add arugula, steak slices, and onions.
Close sandwich and serve hot.