Make the cookies: Whisk together flour, baking soda, and baking powder in a bowl; set aside. Combine butter, sugar, peanut butter, and vanilla paste in a bowl and beat on medium-high speed of a hand mixer until pale and fluffy, about 3–4 minutes. Add egg and beat until smooth. Add dry ingredients, and beat until just combined; stir in oats and peanuts. Transfer to a sheet of plastic wrap, flatten into a disk, and wrap in plastic; refrigerate until firm, about 2 hours.
Transfer to floured work surface, and using a rolling pin, roll until ¼" thick; using a 3 ¼" round cutter, cut out 12 cookies. Using a floured metal spatula, transfer cookies, spaced 2" apart onto parchment paper-lined baking sheets, and freeze until cookie are firm, at least 1 hour. Heat oven to 325°. Bake until golden brown and set, about 16–18 minutes. Let cool completely.
Make the buttercream: Place egg whites in the bowl of a stand mixer fitted with a whisk and beat until soft peaks form; set aside. Bring sugar and ¼ cup water to a boil in a 1-qt. saucepan, stirring to dissolve sugar; attach a candy thermometer to pan, and cook, without stirring, until syrup reaches 248°. Remove from heat, beat egg whites again until stiff peaks form, and while mixer is running, slowly drizzle syrup into whites and continue beating until whites hold stiff peaks and are room temperature, about 15 minutes. Add butter, a few pieces at a time, until buttercream is smooth and emulsified. Measure out 1 ¼ cups and place in a bowl; reserve remaining for another use.
Add peanut butter and salt, and stir until smooth; transfer to a piping bag fitted with a 3⁄8" star tip. Place 6 cookies top side down on a baking sheet, and pipe butter cream evenly over each cookie; top with a remaining plain cookie to create 6 sandwiches. Refrigerate to firm buttercream before serving, at least 30 minutes.