Place the butter in a light-colored saucepan over medium heat. Stir frequently as it melts, foams, and develops golden-brown specks at the bottom. Continue cooking for about 5–8 minutes, until the butter smells nutty and the milk solids turn amber. Immediately remove the pan from the heat and pour the browned butter, including all the flavorful bits, into a heat-safe bowl. Let it cool until slightly warm but no longer hot.
In a large mixing bowl, whisk together the flour, cornstarch, baking soda, and salt. Gently fold in the crispy rice cereal and mini marshmallows, distributing them evenly throughout the dry mixture.
Add the brown sugar, granulated sugar, egg, egg yolk, and vanilla extract to the cooled brown butter. Whisk until everything is well combined and the mixture looks smooth.
Pour the wet mixture into the bowl of dry ingredients. Use a silicone spatula to stir until a thick, slightly crumbly cookie dough forms and no dry flour remains. Gently press the dough together if needed to help it hold its shape.
Scoop the dough into portions of about 2 tablespoons each. Use your hands to firmly shape each portion into a ball, keeping the cereal and marshmallows tucked inside as much as possible. Arrange the dough balls on a parchment-lined baking sheet and freeze them for 1 hour, uncovered, until firm.
Preheat the oven to 350°F (177°C). Line clean baking sheets with parchment paper, then arrange the chilled dough balls about 3 inches apart to allow room for spreading.
Bake for 14–15 minutes, or until the edges are lightly golden and the tops look set while the centers remain slightly soft. If the cookies have not started spreading by around the 10-minute mark, carefully lift the baking sheet and tap it firmly on the counter before returning it to the oven.
Remove the cookies from the oven and sprinkle with flaky sea salt, if desired. Let them rest on the baking sheet for about 10 minutes before transferring them to a wire rack to cool. The edges will become firmer as the cookies cool, while the centers remain pleasantly chewy.