For the Cookies:

1. In a large mixing bowl, cream the butter, shortening, peanut butter, and sugars until light and fluffy.
2. Beat in eggs and vanilla.
3. In a large bowl, combine flour, baking soda and salt.
4. Add to mixing bowl, and mix well.
5. Shape into 1 inch balls and place 2 inches apart on ungreased baking sheets. Flatten slightly with a fork.
6. Bake at 375°F for 7-8 minutes or until golden brown at the edges. Remove to wire racks to cool.

For the Buttercream:

1. In a small saucepan over low heat, combine white chocolate and cream. Stir until chocolate is melted and mixture is smooth.
2. Set aside and let cool to room temperature.
3. Meanwhile, in a large mixing bowl, beat butter until creamy.
4. Gradually add powdered sugar and cream, blending after each addition.
5. Add white chocolate mixture and vanilla, and beat until light and fluffy.
6. Spread the frosting on half of the cookies, and top each with another cookie. Enjoy!