Preheat your oven to 375°F (190°C) and line two large baking sheets with parchment paper.
In the bowl of a stand mixer fitted with the paddle attachment, beat the softened butter, 1 cup of brown sugar, and 1 cup of granulated sugar on medium speed for 2 to 3 minutes until the mixture is light and fluffy.
Add the eggs one at a time, mixing well after each addition until fully incorporated, then mix in the vanilla extract until just combined.
In a separate medium bowl, whisk together the flour, baking soda, baking powder, and salt. With the mixer on low speed, gradually add the dry ingredients to the wet ingredients, mixing just until a soft dough forms and no streaks of flour remain.
Scoop about 2 tablespoons of dough and flatten it in your palm. Place one unwrapped Rolo in the center and carefully wrap the dough around it, sealing it completely into a ball. Roll the ball in the reserved 2 tablespoons of granulated sugar to coat lightly.
Place the dough balls about 2 inches apart on the prepared baking sheets and bake for 9 to 11 minutes, or until the edges are just set and the tops look slightly cracked. The centers will still appear soft.
Let the cookies cool on the baking sheet for 5 minutes to allow them to set, then transfer them to a wire rack to cool completely.