Caramelise the Onions: Heat the olive oil and 1 tbsp of butter in a large frying pan over a medium-low heat. Add the sliced onions and cook slowly for 20-25 minutes, stirring occasionally, until they are very soft, sweet, and deep golden brown. For a detailed guide on this technique, Serious Eats has a fantastic tutorial.

Deglaze the Pan: Once the onions are caramelised, stir in the brown sugar and balsamic vinegar. Cook for another minute until the vinegar has reduced and coated the onions. Remove the onions from the pan and set them aside.

Make the Horseradish Sauce: While the onions are cooking, prepare the sauce. In a small bowl, combine the crème fraîche, mayonnaise, horseradish cream, Dijon mustard, chopped chives, and lemon juice. Season with a pinch of salt and pepper. Stir well and set aside.

Toast the Buns: Wipe out the frying pan. Spread the 2 tbsp of softened butter on the cut sides of the brioche buns. Place them cut-side down in the warm pan and toast for 1-2 minutes until golden brown and slightly crisp. Remove and set aside.

Warm the Beef: Place the same frying pan back over a medium heat. Add the sliced smoked beef to the pan, spreading it out. Heat for 2-3 minutes, turning it gently with tongs, until it's warmed through.

Melt the Cheese: Divide the warmed beef into four equal portions in the pan. Place two slices of Provolone cheese over each portion of beef. I find that placing a lid over the pan for 30-60 seconds helps the cheese get gloriously melty and oozy.

Assemble Your Sandwiches: Spread a generous layer of the horseradish sauce on both the top and bottom toasted buns. Using a spatula, carefully lift each cheesy beef pile onto a bottom bun.

Finishing Touches: Top the beef and cheese with a good helping of the sweet caramelised onions. Place the top bun on, press down gently, and serve immediately while everything is warm.