Preheat the oven to 350 degrees. Prepare two large rimmed baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking powder, baking soda, and salt.
Beat the butter and medium speed for 5 minutes. Scrape down the sides. Add the sugar and continue beating for 5 to 7 minutes or until fluffy.
Add the egg and vanilla extract, beating until evenly combined. Add the flour mixture and beat until mixed.
Beat ½ cup of the graham cracker crumbs into the batter.
Using a small (1 tablespoon) cookie scoop, portion the cookie dough and shape into balls. Roll each ball in the remaining graham cracker crumbs. Place on the baking sheet about 1 inch apart.
Bake for 8 to 10 minutes or until the cookies are puffed and the edges are set.
Remove the baking sheets and allow the cookies to rest while the broiler preheats.
Preheat the oven broiler on high.
Place one-half marshmallow on each cookie. Put the cookies on the oven rack beneath the broiler with about touching and allow the marshmallows to toast until golden, about 1 minute.
Remove the cookie sheets from the oven and immediately press one candy into the center of each marshmallow.
Transfer to a wire rack and allow the cookies to cool completely. The candies will look like they are melting but do not disturb them and they will resolidify in the correct shape.
Store in an airtight container for up to 1 week or refrigerate for up to 3 weeks. Cooled cookies can also be frozen but sudden temperature changes may cause the candy to “bloom” which is harmless but not pretty.