Prepare the Setup: Place the steamer rack or trivet in the bottom of your cold skillet. Add the 1/4 cup of water to the skillet, ensuring the water level is below the rack.
Form the Patties: Gently divide the beef into 4 equal portions (about 4 oz / 115g each). Loosely form into balls, then gently flatten into 3/4-inch thick patties. Make a slight dimple in the center of each with your thumb to prevent puffing. Season both sides generously with salt and pepper.
Steam the Burgers: Place the patties on the steamer rack, ensuring they are not sitting in the water. Place your empty ramekin or bowl in the center of the skillet among the patties. This will act as a platform for your cheese.
Cover and Cook: Set the skillet over medium-high heat. Once the water begins to simmer and produce steam, cover the skillet tightly with the lid. Steam the burgers for 6-8 minutes for medium doneness (135-140°F / 57-60°C). Avoid lifting the lid frequently, as this releases steam.
Melt the Cheese: In the last 2 minutes of cooking, place the cheese slices into the ramekin in the center of the skillet. The steam will gently melt it into a gooey, velvety pool.
Steam the Buns (Optional): In the final minute, you can place the halved buns, cut-side down, on top of the patties or around the edge of the skillet to warm and soften them.
Assemble: Carefully remove the lid. Use the spatula to place a patty on the bottom bun. Spoon a generous amount of the melted cheese from the ramekin over the patty. Add your desired toppings and condiments, top with the other bun half, and serve immediately.