Grease a 9×13-inch baking dish with nonstick spray or a little butter to ensure easy removal later.
In a large stainless steel pan, melt the butter over medium-low heat. As the butter heats, it will foam and sizzle. Keep the butter moving by swirling the pan occasionally. When the crackling noise subsides, continue cooking until you notice the butter turning golden brown, releasing a warm, nutty fragrance, and dark amber bits collecting at the bottom of the pan. This process should take around 2 to 3 minutes after the sizzling stops. At this point, reduce the heat to low and stir in the marshmallows, vanilla extract, and kosher salt. Keep stirring until the marshmallows are fully melted and the mixture is smooth.
Once melted, remove the pan from heat and fold in the crispy rice cereal. Stir gently but thoroughly until all the cereal is evenly coated in the marshmallow mixture. Transfer this mixture to your prepared baking dish and gently press it into an even layer using a spatula, wooden spoon, or your damp hands. Be careful not to press too hard—you want the treats to stick together but remain chewy rather than dense.
Let the rice krispie treats cool completely at room temperature, about 1 hour. Once set, slice them into 12 squares. Store them in an airtight container at room temperature for up to 2 days.