Sift the all-purpose flour, baking soda, salt, and chopped milk chocolate into a medium mixing bowl.
Set the flour mixture aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat the softened butter, granulated sugar, and dark brown sugar until light and fluffy.
Scrape down the sides of the bowl as needed.
Add the eggs, one at a time, mixing until each egg is fully incorporated before adding the next.
Add the vanilla extract and creamy peanut butter.
Beat until just incorporated.
Add half of the flour mixture to the butter mixture.
Mix for approximately 15 seconds.
Add the remaining flour mixture.
Mix only until the dough is just combined.
Cover the bowl with plastic wrap.
Refrigerate the dough for at least 3 hours.
Preheat the oven to 375°F.
Line a baking sheet with parchment paper, if desired.
Drop the chilled dough by rounded teaspoonfuls onto the baking sheet, spacing each portion at least 2 inches apart.
Using the palm of your hand, gently press each portion into a tall disk.
Do not flatten the cookies too much.
Sprinkle the tops lightly with granulated sugar.
Bake for approximately 8 minutes, or until the tops of the cookies just begin to brown.
Remove the baking sheet from the oven.
Allow the cookies to cool on the baking sheet for 2 minutes.
Transfer the cookies to a wire cooling rack.
Allow the cookies to cool completely before filling.
To prepare the filling, place the chopped milk chocolate, creamy peanut butter, powdered sugar, and coarse kosher salt in a medium bowl.
Set aside.
In a small heavy saucepan, bring the whipping cream to a boil over medium heat.
Pour the hot cream over the chocolate and peanut butter mixture.
Stir until the mixture is fully melted, smooth, and evenly combined.
Transfer the filling to the refrigerator.
Chill for approximately 1 hour, or until thick and spreadable.
Spread about 1 rounded teaspoonful of the milk chocolate peanut butter filling onto the flat side of one cooled cookie.
Top with a second cookie to form a sandwich.
Press gently to secure the filling.
Repeat with the remaining cookies and filling.
Store the finished sandwich cookies in an airtight container.