Step 1: In a small bowl, combine mayonnaise, sour cream, prepared horseradish, minced chives (if using), and lemon juice. Season with salt and black pepper. Stir until well combined. Cover and refrigerate to allow flavors to meld.
Step 2: Heat olive oil in a large skillet or frying pan over medium-low heat. Add the thinly sliced onions and a pinch of salt. Cook, stirring occasionally, for 20-30 minutes, or until the onions are deeply golden brown and very tender and sweet. Be patient; this step is key.
Step 3: While the onions are caramelizing, gently warm your prime rib slices. You can either add them directly to the pan with the onions for the last 5 minutes of cooking, or lightly sear them in a separate hot pan for 1-2 minutes per side until just warmed through. Be careful not to overcook.
Step 4: Lightly butter one side of each bread slice. In a separate clean pan or on a griddle, toast the bread over medium heat until golden brown and crispy.
Step 5: Once toasted, spread a generous layer of horseradish cream on the unbuttered side of each slice of bread.
Step 6: Pile a desired amount of warm prime rib onto four slices of bread, followed by a generous spoonful of caramelized onions. Top with the remaining bread slices, horseradish cream side down.
Step 7: Slice the sandwiches in half, if desired, and serve immediately. Serve with a side of au jus for dipping if available.