Using a mixer or by hand, cream together butter, peanut butter, and both sugars.
Add egg and vanilla extract to the creamed butter and sugar, and mix until smooth.
In a large bowl, whisk together flour, salt, baking powder, and baking soda. Add flour mixture to wet ingredients, about a third at a time, until dry ingredients are fully incorporated.
Chill dough in fridge for at least 30 minutes.
Preheat oven to 375°F. Line a baking sheet with parchment paper or Silpat.
Using a spoon or cookie scoop, scoop the dough into 1 tablespoon balls. Place about 2" apart on the baking sheet.
Bake for 10 minutes. Remove cookies from the oven, and carefully press 3-4 marshmallows into each cookie. Return to the oven and bake another 2-3 minutes, until marshmallows have puffed up and start to get melty.
Sprinkle sea salt on cookies while hot. Let cool on baking sheet for 5 minutes, then transfer to a cooling rack and let cool completely.