Brown the butter: Cut the butter into pieces and place it in a light-colored skillet or saucepan over medium heat. As the butter melts, it will begin to bubble and foam. Continue cooking, stirring frequently, as the foam starts to subside and golden-brown specks begin to form on the bottom of the pan.
Keep stirring the butter and watch closely as the butter can go from perfectly browned to burnt quickly. Once the butter is golden brown and smells rich and nutty, immediately remove it from the heat and pour it, including all of the browned bits, into a large heatproof mixing bowl. Let the butter cool slightly for about 15 minutes before continuing with the recipe.
Make the cookie dough: Add the brown sugar, granulated sugar, eggs, and vanilla to the slightly cooled brown butter. Whisk until smooth and well combined.
Add the flour, baking soda, cornstarch, and salt. Mix just until the dry ingredients are incorporated. Gently fold in the Rice Krispies cereal, being careful not to crush them.
Scoop and add the frozen marshmallows: Using a 2-Tablespoon cookie scoop, scoop the dough and flatten it slightly. Place about 5 frozen mini marshmallows in the center of each piece of dough. Wrap the dough around the marshmallows, leaving a small opening at the top so the marshmallows can peek through as they bake. Place the cookie dough balls on a parchment-lined baking sheet or plate.*Keep the marshmallows in the freezer between batches so they stay completely frozen. This helps them hold their shape while the cookies bake. Chill the cookie dough balls: Once all of the cookies have been assembled, refrigerate the cookie dough balls for at least 2 hours. You can also chill them overnight.
Bake: When ready to bake, preheat the oven to 325°F. Line baking sheets with parchment paper or silicone baking mats and place only 6 chilled cookie dough balls on the prepared sheets, leaving lots of space between each cookie.
Bake for 12 minutes, or just until the edges begin to turn golden. The centers may look slightly underbaked and that is fine. The cookies will continue to set as they cool on the baking sheet. Keep in mind that oven temperatures can vary, so watch them closely.
Shape the cookies: If needed, use a round cookie cutter or biscuit cutter to gently scoot the edges of the warm cookies into a perfect circle. Let the cookies cool on the baking sheet for about 5 minutes, then transfer them to a wire cooling rack. Garnish: Sprinkle the warm cookies with flaky sea salt, then serve and enjoy!