Preheat oven to 350°F. Line two cookie sheets with parchment paper or Silpat baking mats.

In the bowl of an electric mixer fitted with the paddle attachment (or in a large bowl with a hand mixer), cream 1/2 cup butter (softened), 3/4 cup peanut butter, 3/4 cup packed brown sugar, and 1/4 cup granulated sugar until light and combined.

Add 1 large egg, 1 teaspoon vanilla, 1 tablespoon milk, 1/4 teaspoon salt, and 1/2 teaspoon baking soda; mix until incorporated.

Gradually add 1 3/4 cups all-purpose flour and mix on low speed until the dough comes together and no dry flour remains.

Use a tablespoon to scoop dough into 1-tablespoon balls. Place the balls on the prepared cookie sheets about 2 inches apart.

Press each dough ball down with the tines of a fork to create a criss-cross pattern.

Bake for 8–10 minutes, until the cookies just begin to brown at the edges. Allow the cookies to cool completely on the cookie sheets (they will firm up as they cool).

Make the filling: in a medium bowl, beat 4 tablespoons unsalted butter (softened) and 1/2 cup creamy peanut butter until smooth.

With the mixer on low, add 2 cups powdered sugar in batches until incorporated and the mixture is thick.

Add 2 teaspoons vanilla extract and 1 teaspoon maple extract and mix to combine.

Add 2 tablespoons heavy whipping cream, then add additional heavy whipping cream 1 tablespoon at a time (up to a total of 3–5 tablespoons) until the filling reaches a spreadable consistency.

Turn half the cooled cookies upside down, spread an even amount of filling on each, and sandwich with the remaining cookies. Store in an airtight container for up to 3 days.