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.
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.
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.
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.