Banana Pudding Cookies

1 cup salted butter softened
¾ cup light brown sugar
¼ cup white granulated sugar
1½ teaspoons real vanilla extract
2 large eggs
2⅓ cups all purpose flour
1 package banana flavored instant pudding mix, 2 tablespoons reserved for the frosting (see below) 5.1 ounce package
1 teaspoon baking powder
1 teaspoon baking soda
optional: 1 cup white chocolate chips, 1 cup crushed walnuts