Step 1

Preheat oven to 350°F. Whisk first 4 ingredients in medium bowl. Using electric mixer, beat powdered sugar, dark brown sugar, and butter in large bowl to blend. Add peanut butter; beat until creamy. Gradually beat in vegetable oil and vanilla extract, then egg. Add dry ingredients; mix just until blended. Stir in milk chocolate chips.

Step 2

Drop cookie dough by level tablespoonfuls onto ungreased baking sheets, spacing about 1 1/2 inches apart. Bake cookies until puffed and golden brown, about 12 minutes. Cool slightly, then transfer cookies to rack to cool completely.

Step 3

Place chocolate, peanut butter, powdered sugar, and kosher salt in medium bowl. Bring whipping cream to boil in heavy small saucepan. Pour hot cream over chocolate mixture; stir until mixture is melted and smooth. Chill until filling is thick and spreadable, about 1 hour.

Step 4

Spread about 1 rounded teaspoonful chocolate-peanut butter filling on flat side of 1 cookie. Top with second cookie, forming sandwich. Repeat with remaining filling and cookies. DO AHEAD:Cookie sandwiches can be made 1 day ahead. Store in airtight container at room temperature.