Step 1

Preheat the oven to 450 degrees F.

Step 2

Heat the olive oil in a medium Dutch oven or heavy-bottomed pot over medium heat. Add the onion and cook, stirring, until softened, about 4 minutes. Add 2 tablespoons butter; when the butter has melted, add the mushrooms and thyme. Cook, stirring occasionally, until the mushrooms soften and start to caramelize, 4 to 5 minutes.

Step 3

Meanwhile, in a small skillet over medium heat, melt the remaining 4 tablespoons butter. Add the garlic and cook, stirring occasionally, until the garlic is fragrant, about 3 minutes. Place the rolls cut-side up on a rimmed baking sheet. Liberally brush the rolls with the garlic butter and sprinkle with salt and pepper. Toast in the oven until golden brown, 4 to 5 minutes. Set aside for assembling.

Step 4

To the pot with the mushrooms, add the beef broth and bring to a simmer. Taste and adjust the seasoning as necessary. Bring the heat down to medium-low and simmer an additional minute. Turn the heat off, then add the roast beef to the pot, quickly removing when just warmed through, about 30 seconds.

Step 5

To assemble the sandwiches, spread a layer of creamy horseradish, to taste, on both sides of the rolls. Add a layer of beef on the bottom halves of each roll, then top with a spoonful of mushrooms along with some of the jus. Top with giardiniera to taste. Transfer the remaining jus to dipping bowls and serve on the side.