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Preheat your oven to 350°F (175°C). Lightly grease a 9×13-inch glass baking dish so the rolls don’t stick.
Lay the thin beef slices out on a cutting board. If any pieces are very large, cut them in half so you have 8–10 roll-sized pieces. Trim any thick, tough edges if needed so they roll more easily.
Place one slice of Swiss cheese on each piece of steak, tearing the cheese to fit if needed so most of the meat is covered.
Sprinkle a small spoonful of the drained sliced mushrooms along one short end of each cheese-covered steak piece, dividing the mushrooms fairly evenly among all the pieces.
Starting at the end with the mushrooms, roll each steak up tightly around the filling, like a little bundle, keeping the cheese and mushrooms tucked inside. Place each roll seam-side down in the prepared glass baking dish, snugging them close together so they hold their shape.
In a medium bowl, whisk or stir the condensed brown gravy until smooth. (If it’s very thick, you can add just a splash of water to loosen, but not so much that it stops being a rich gravy.)
Pour the gravy evenly over the steak rolls, making sure each roll is well coated and there’s gravy in the bottom of the dish. Use a spoon to nudge gravy between the rolls if needed.
Cover the baking dish tightly with foil to keep the rolls moist. Bake in the preheated oven for 45 to 60 minutes, or until the beef is tender when pierced with a fork and the cheese inside is melted.
Carefully remove the foil, then bake uncovered for an additional 5 to 10 minutes to slightly thicken and deepen the color of the gravy on top.
Let the steak rolls rest for about 5 minutes before serving. Spoon extra gravy from the bottom of the dish over each serving, making sure everyone gets some of the mushrooms and melted Swiss tucked inside.
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