In a medium bowl, whisk together the chicken broth, steak sauce, and Dijon mustard until well combined. Set aside.
Heat a large 12-inch skillet over medium-high heat. Add the ground beef and cook for 7 to 8 minutes, breaking it apart with a spatula, until fully browned. Drain excess grease and transfer the beef to a plate.
In the same skillet, melt the butter over medium heat. Add the sliced mushrooms, chopped onion, and Montreal steak seasoning. Sauté for 7 to 8 minutes, or until the onions are translucent and the mushrooms are golden brown.
Sprinkle the flour evenly over the vegetables in the pan. Cook for 1 to 2 minutes, stirring constantly, until the flour smells nutty and is well incorporated.
Return the cooked ground beef to the skillet. Pour in the broth mixture and stir to combine.
Bring the mixture to a simmer and cook for 3 to 4 minutes, stirring occasionally, until the sauce has thickened significantly.
Spoon the hot beef and mushroom mixture onto toasted bun bottoms. Top immediately with shredded cheese so it melts, then cover with the top bun and serve.