Coat the bottom of a large skillet with 1 tablespoon of oil and place over medium-low heat. Add onion and toss a few times to coat with oil. Allow to cook until caramelized, about 20 minutes, flipping occasionally. Add garlic and cook about 2 minutes more. Transfer onions and garlic to a plate.

Coat skillet with another tablespoon of oil and raise heat to medium. Add mushroom strips. Avoid overcrowding the skillet. A little overlapping is okay, but work in multiple batches if needed (I needed two), adding a bit of oil between batches if needed. Cook until lightly browned, about 5 minutes. Flip and cook 5 minutes more on opposite sides.

Return onions to skillet and add broth, soy sauce, Worchestershire sauce, thyme, liquid smoke and pepper. Bring to a simmer and allow to cook, stirring occasionally, until liquid is reduced by half, about 5 minutes.

Slather the insides of rolls with horseradish mustard. Use a slotted spoon to remove onions and mushrooms from skillet, pressing lightly to squeeze out any excess juice. Divide onions and mushrooms into rolls. Pour cooking liquid into a small bowl and serve with sandwiches, for dipping.