1. Caramelize the Onions: In a pan, melt butter over medium-high heat. Add the thinly sliced onions and let them get brown on the bottom before turning. Reduce heat to medium-low and simmer until they become soft and caramelized. This process can take 20-30 minutes. Season with a pinch of sugar (optional), salt, and pepper to enhance the caramelization process. Stir occasionally.
2. Prepare the Horseradish Sauce: In a small bowl, mix horseradish sauce and mayonnaise. Adjust the ratio to your preferred level of spiciness. Set aside.
3. Toss the Salad: In a small bowl, toss the arugula with the olive oil and lemon juice until well coated. Set aside.

Assemble the Sandwich:

1. Preheat a grill pan or skillet over medium heat. Alternatively, you can use a toaster or oven to toast the bread. Butter one side of each slice of bread.
2. Spread the horseradish-mayo sauce on the unbuttered side of one slice of bread.
3. Layer the thinly sliced prime rib and provolone or Swiss cheese on top (if using).
4. On the other slice of bread, spread the caramelized onions.
5. Put the two slices of bread together with the spreads and fillings inside.
6. Cut the sandwich in half, and serve it with your favorite sides like coleslaw, pickles, or potato chips.

Serve with Optional Au Jus:

1. In a small saucepan, heat the reserved juices from the cooked prime rib. If needed, add a bit of beef broth to adjust the consistency. Season the au jus with salt and pepper to taste.
2. Serve the sandwich with a side of the optional au jus for dipping.