Melt the butter in a skillet over medium-low heat. Continue to cook until the mixture bubbles and the milk solids brown. Transfer the browned butter to a large bowl and let cool completely until solid but still soft. (You're aiming for softened butter texture.)
Mix together the flour and baking soda in a medium bowl and set aside. Beat the cooled browned butter and sugars with a hand mixer until creamy.
Add the eggs to the butter mixture one at a time, beating just until the yellow disappears after each addition. Beat in the vanilla. Add the flour mixture and beat until smooth.
Stir in the chocolate chips and pretzels. Scoop the dough into 48 equal balls and place on parchment paper-lined baking sheets. Press the dough into discs.
Top 24 of the discs with a marshmallow half. Place the remaining dough discs on top of the marshmallows.
Roll the dough into balls completely covering the marshmallow inside. Refrigerate for 1 hour.
Preheat the oven to 350°F. (Optional: Press a few extra chocolate chips and pretzel pieces into each dough ball.) Bake the cookies, in batches, 11 to 13 minutes or until golden brown. (Optional: During the last 2 minutes of baking, add a marshmallow piece to the top of the cookies.) Remove the cookies from the oven and let cool completely on the baking pans.