Preheat oven to 350°F. Line baking sheets with parchment.

Whisk flour, baking soda and salt in a bowl.

Beat butter, brown sugar and 1 cup peanut butter until light and fluffy.

Beat in egg and 1/2 tsp vanilla.

Gradually mix in dry ingredients until a dough forms.

Roll dough into 1-inch balls, then into logs. Coat in granulated sugar.

Place logs 2 inches apart on baking sheets. Use a fork to make criss-cross pattern and pinch into peanut shapes.

Bake 10-12 minutes until lightly browned on edges. Cool 5 mins on sheets, then transfer to racks to cool completely.

Beat 1/4 cup butter until smooth and creamy.

Mix in 1/2 cup peanut butter, then 1 cup powdered sugar, milk and 1/4 tsp vanilla until combined.

Gradually mix in remaining 1/2 cup powdered sugar until smooth and spreadable.

Spread 1 tbsp filling on the flat side of one cookie, top with another cookie to make sandwiches.