In a large bowl, beat together butter, brown sugar, and caster sugar until light and fluffy. Beat in the egg, then the vanilla extract.
Mix in the flour, baking powder, baking soda, and salt. Fold in the white chocolate chips. Cover and refrigerate 2 hours to overnight.
Preheat oven to 350 degrees F (180 C) and line 2 baking sheets with parchment or lightly grease. Roll each cookie into the size of a golf ball and place on prepared baking sheets, 2 inches apart. Bake in preheated oven until golden on the bottom, 7-8 minutes. Leave cookies on baking sheet for 10 minutes before removing to wire rack.
As the cookies cool, prepare drizzle by placing peanut butter and white chocolate in a small bowl. Microwave in 20 second increments, stirring between, until smooth. Drizzle over the top of cooled cookies. The topping will only set slightly.