Add butter and both sugars to a stand mixer fitted with a paddle attachment. Cream until light and fluffy.
While the butter is creaming, whisk together the flour, baking soda, and the salt in a medium-sized bowl.
Add the eggs, and the vanilla to the butter and sugar mixture. Once the eggs and vanilla are added and smooth, add the dry ingredients in 1/2 cup increments until just incorporated. Scrape down the sides of the mixing bowl with a rubber spatula, making sure everything is mixed together evenly. Stir in the chocolate chips.
Drop heaping tablespoons of cookie batter onto 2 sheet trays lined with parchment, leaving about 2-inches of space between each cookie. Bake in the preheated oven for 13 to 15 minutes. Remove and let cool completely on the cookie sheets.
In a large mixing bowl, beat the peanut butter until smooth. Add butter and continue beating until thoroughly mixed and smooth. Add the confectioners' sugar in 1/2 cup increments and mix until combined.
Spread a heaping tablespoon of frosting onto the flat side of a cookie. Top with another cookie. Repeat with remaining cookies and filling. Arrange on serving dish and serve.