Mix white sugar, evaporated milk, brown sugar, and butter in a saucepan; bring to a boil. Reduce heat to a simmer and cook, stirring continually, until mixture is smooth and sugar is dissolved, about 8 minutes. Remove saucepan from heat and stir marshmallow creme into sugar mixture until smooth.
Stir cookie butter into sugar-marshmallow mixture until evenly combined; pour into the prepared baking dish. Let fudge stand until cooled and solid, at least 30 minutes.