In a medium bowl combine prepared horseradish, sour cream, mayonnaise, lemon juice, Dijon mustard and chopped chives. Season with 1/4 tsp salt and 1/8 tsp black pepper. Taste and adjust heat with more horseradish or lemon if desired. Cover and chill at least 30 minutes to meld flavors (can be made up to 24 hours ahead).

House Au Jus

Sauté aromatics

In a medium saucepan over medium heat melt 1 tablespoon butter. Add the minced shallot and smashed garlic cloves and sauté until softened and lightly golden, about 2–3 minutes, stirring so they don't burn.

Deglaze and simmer

Add optional red wine to the pan and simmer 1–2 minutes to reduce slightly, scraping any browned bits. Pour in beef broth and add the beef demi-glace, thyme sprigs and bay leaf. Bring to a gentle simmer, reduce heat to low and simmer uncovered 10–12 minutes to concentrate flavor.

Finish and strain

Remove thyme sprigs and bay leaf. Taste and season with 1/2 tsp salt and 1/4 tsp black pepper (adjust to taste). Strain the jus through a fine-mesh sieve into a warm serving vessel to remove shallot and garlic solids. Keep warm; reheat gently before serving.

Beef & Sandwich

Prep beef

If not pre-shaved, freeze the New York strip for 30–45 minutes until firm but not rock solid, then using a sharp knife slice very thin across the grain to yield about 1 lb thin shaved beef. Season the shaved beef lightly with 1 tsp kosher salt and 1/2 tsp freshly ground black pepper; toss to distribute.

Quick sear

Heat a large skillet or griddle over high heat until very hot. Add 1 tablespoon vegetable oil and 1 tablespoon butter. Add shaved beef in a single even layer (work in batches if necessary) and sear just until browned but still juicy, about 30–45 seconds per side. Remove beef to a plate and keep warm; reserve pan juices.

Toast rolls & assemble

Slice rolls lengthwise (keep hinge if desired). Spread remaining 1 tablespoon butter on cut sides. Return skillet to medium heat and toast cut sides until golden, 1–2 minutes. Pile an equal portion of the hot shaved beef onto the bottom half of each roll, then top each with 2 oz sliced Gruyere. Place assembled sandwiches under a broiler or in a 375°F (190°C) oven just until the cheese melts, about 2–4 minutes—watch carefully so bread doesn't burn.

Serve

Transfer sandwiches to plates. Serve with a small ramekin of warm house au jus for dipping and a dollop (about 2 tablespoons) of chilled horseradish cream on the side. Provide extra au jus and horseradish cream to taste.