To make the peanut butter filling, combine all of the filling ingredients in a bowl using a whisk. Cover and chill until you are ready to use.
To make the peanut butter cookies: in a bowl, sift together the flour, baking soda, and baking powder. Stir in the oats and salt; set aside.
In the bowl of an electric mixer fitted with the paddle attachment, combine the butter, chunky peanut butter and sugars and mix on medium-high speed until very fluffy and pale, for at least 3 minutes, scraping down the bowl as needed.
Turn the mixer to medium-low and add the eggs, one at a time, beating to incorporate each egg and scraping down the bowl as needed. Beat in the vanilla extract and then add the dry ingredients on low speed in 3-4 additions and mix until just combined. Do not overmix. Scrape down the bowl with a rubber spatula to ensure there is no flour mixture left on the bottom.
Use an ice cream scoop to portion all the cookies in 1 ounce scoops (or use about 1 heaping tablespoon per cookie), placing the dough balls on a parchment paper lined cookie sheet. Tom says you should end up with about 48 cookies but I only had about 40 (using about 1 heaping tablespoon per cookie dough ball). (You can place all the dough balls on one cookie sheet during the chilling step- you will space these out later when you bake them). Chill the scooped dough balls for at least 2 hours or longer.
When ready to bake, preheat your oven to 350 degrees (See my note in the summary above). Arrange 8 cookie dough balls, spaced evenly apart and staggered, on a parchment-lined baking sheet. Bake until evenly golden, for about 12 minutes, rotating the pan halfway through the baking time. Remove from the oven and let the cookies cool completely before filling them.
To make the cookie sandwich: turn one cookie flat and spread with a little less than 2 teaspoons of filling. Top with another cookie, flat side down, pressing gently. Repeat with all other cookies.