Add the butter to a small saucepan over medium heat. Cook, stirring frequently, as the butter melts, foams, and begins to brown. Once the milk solids at the bottom of the pan are golden brown and the butter smells nutty, about 5–7 minutes, immediately remove it from the heat. Pour the browned butter, including all of the browned milk solids, into a large mixing bowl and let cool slightly.
Add the brown sugar and granulated sugar to the browned butter and whisk until well combined. Add the egg, egg yolk, vanilla extract, and heavy cream and whisk until smooth.
In a separate medium bowl, whisk together the flour, cornstarch, baking soda, baking powder, and salt.
Add the dry ingredients to the wet ingredients and fold with a rubber spatula just until combined. Fold in the dark chocolate chunks and milk chocolate chips, being careful not to overmix.
Cover and refrigerate the cookie dough for at least 2 hours, or until thoroughly chilled and firm. Do not skip the chill time. The cold dough is much easier to wrap around the s’mores centers and helps control spreading during baking.
When ready to bake, preheat the oven to 365°F.
Break the 4 whole graham cracker sheets in half to make 8 squares. Top each graham cracker square with 1 large marshmallow and 3–4 chocolate melting wafers. You may only need 7 centers depending on your final cookie yield.
Take a portion of chilled cookie dough and flatten it into a thick disc in your hands. Place one prepared s’mores center in the middle, then bring the cookie dough up and domed around the graham cracker, marshmallow, and chocolate none of the filling is exposed.
Weigh each fully assembled cookie, including the s’mores filling, and aim for approximately 120g per cookie. Add or remove a little cookie dough as needed. Depending on your exact measurements, the recipe will make about 7–8 large cookies.
Arrange the assembled cookies on the baking sheet, leaving plenty of room between them to spread. Bake for 11–12 minutes, until the edges are golden and set but the centers still look soft and slightly underbaked.
Immediately after removing the cookies from the oven, use a large round cookie cutter to gently scoot around each cookie and reshape it into a circle, if desired. Press a few additional milk chocolate chips into the tops while the cookies are still warm.
Let the cookies cool on the baking sheet for about 5 minutes before transferring them to a wire cooling rack. The cookies will continue to set as they cool.
