Place the shallots and oil in small saucepot and cook over high heat until the oil starts to bubble. Reduce the heat to medium and continue to cook, stirring frequently and adjusting the heat to maintain a steady stream of bubbles. Cook until the shallots are rust colored, about 15 minutes. Carefully pour through a strainer into a second pot, then transfer the shallots to a paper-towel-lined plate. (Save the oil for stir-fries or for second batch of shallots.) Separate the shallots with a fork and season with salt. Allow to cool completely before storing in a sealed container (will keep for several weeks at room temperature).

Sprinkle the beef with 3 tablespoons salt. Set on a rack in a baking dish, wrap in plastic, and refrigerate overnight.

The next day, preheat the oven to 475 degrees F and adjust the rack to the lower-middle positions. Roast the beef until it is lightly browned, 35 to 45 minutes. Flip the roast, reduce the heat to 300 degrees F and continue roasting until the center registers 135 degrees F on an instant-read thermometer, 45 minutes to 1 hour longer. Remove from the oven and allow to rest at least 45 minutes. Slice as thinly as possible.

Combine the thousand island and horseradish in a small bowl. Split the rolls in half and toast. Spread the dressing generously on the cut sides. Sprinkle the shallots generously on the top side of each roll. Top with the cheese. Place about 4 ounces beef on the bottom half of each sandwich. Close the sandwiches, cut and serve.