Preheat oven to 375°F (190°C). Slice each French roll horizontally, leaving a small hinge if desired. In a small bowl, mix softened butter, Dijon mustard, horseradish, salt, and black pepper until combined. Spread the butter mixture evenly on cut faces of each roll.

Assemble protein and cheese

Warm the sliced prime rib gently in a skillet over low heat for 2–3 minutes with a splash (about 1 tbs) of beef broth to prevent drying, just until warmed through. Pile about 4 oz (¼ lb) of warmed prime rib onto the bottom half of each roll, then top each with about ½ cup shredded sharp cheddar (divide evenly so total equals 2 cups).

Bake to melt cheese

Close sandwiches and place on a baking sheet. Bake in preheated oven for 8–10 minutes, then switch oven to broil for 1–2 minutes (watch closely) to brown cheese and toast tops. Alternatively, toast under broiler for 2–3 minutes until cheese is melted and edges are golden.

Finish

Remove sandwiches from oven and rest 1–2 minutes. Open slightly and, if desired, add a few fresh thyme leaves. Serve each sandwich with a ramekin of warm au jus for dipping.

Au Jus

Sauté aromatics

In a small saucepan over medium heat, melt 1 tablespoon butter. Add minced garlic and minced shallot; sauté 1–2 minutes until fragrant and softened but not browned.

Deglaze and combine

Add dry red wine (if using) and scrape any browned bits from pan. Let wine reduce 1 minute. Stir in demi-glace and beef broth, then add Worcestershire sauce, soy sauce, and the thyme sprig. Bring to a gentle simmer for 4–5 minutes to meld flavors.

Adjust thickness and finish

If you prefer a slightly thicker jus for dipping, whisk cornstarch with cold water to make a slurry and stir into simmering jus; simmer 1 minute until slightly thickened. Taste and adjust seasoning with a pinch of salt or a dash more Worcestershire if desired. Strain through a fine-mesh sieve into a warm serving bowl or individual ramekins, discarding solids and thyme sprig. Keep warm until serving.