Combine the butter and sugars in a large mixing bowl and mix together. Add in egg and vanilla and mix until just combined.
Slowly mix in flour, salt, baking soda, and ¼ c. graham crackers.
Scoop into 1½” balls, and refrigerate for 15-20 mins up to overnight.
Unwrap the kisses and form roughly 1” pillows of marshmallows by stretching them slightly so they stick together. Alternatively, cut 10 large marshmallows in half. Line a baking sheet with parchment paper.
Roll each dough into remaining graham crackers until covered. Bake 8-10 minutes or until slightly underbaked, with toasted bottoms.
Remove the pan from the oven and turn temperature up to 450°F. Place marshmallows on top of each cookie and bake 1-2 more minutes.
Remove pan when toasted, and press kisses onto top while warm.
Allow to cool completely before storing. Warm in microwave for 10 seconds to enjoy warm after cooling.