Heat a 10-inch skillet over medium heat. Add olive oil and when hot, add the thinly sliced onion rings. Sprinkle with 1/4 teaspoon kosher salt and cook, stirring occasionally, until they are softened and just beginning to caramelize, about 8–10 minutes. Remove from heat and set aside.

Prepare the rolls

Slice each Kaiser roll horizontally. Spread about 1/2 tablespoon of softened butter on the cut face of each roll (both top and bottom). Lightly toast the buttered rolls on a grill pan or skillet over medium heat until golden and crisp, about 1–2 minutes per side. Remove and set top halves aside.

Warm the roast beef

In the same skillet used for onions (wipe out any excess if necessary), lower heat to medium-low. Add roast beef in a single layer, sprinkle with remaining 3/4 teaspoon kosher salt, 1/2 teaspoon black pepper, and the optional Worcestershire sauce; toss gently. Heat just until beef is warmed through, 2–3 minutes. For ease, divide the beef into four equal portions (about 4 oz each) while warming.

Assemble sandwiches

Place warmed bottom halves of the rolls on a baking sheet or ovenproof skillet. Divide the warmed roast beef evenly onto each bottom roll. Top the beef with about 2 ounces (2 slices) of sharp cheddar per sandwich, then evenly distribute the cooked onions over the cheese. Spread approximately 2 tablespoons of prepared horseradish onto each top roll (adjust to taste), then place the tops over the onions/cheese.

Melt the cheese

Preheat broiler on high (or oven to 450°F/230°C). Place the assembled sandwiches under the broiler for 1–3 minutes, just until the cheddar melts and begins to bubble—watch carefully to avoid burning. Alternatively, place the sheet in the preheated oven for 4–6 minutes until cheese is melted.

Finish and serve

Remove sandwiches from the oven/broiler. Let rest 1 minute. If desired, give a quick press with a spatula for a minute to meld components. Serve each Englishman sandwich warm. Offer extra horseradish on the side.