Preheat the oven to 350 degrees Fahrenheit. Prepare a baking sheet with parchment paper and set aside.
Combine the softened butter and brown sugar in a mixing bowl. Beat together until creamy.
Add the eggs and vanilla, mixing until combined.
Slowly fold in the flour, crushed graham crackers, salt, baking soda, and baking powder using a rubber spatula until incorporated and no dry-ingredient streaks are left.
Use a large cookie scoop to make 2-inch round portions of cookie dough and place them on the prepared baking sheet about 2 inches apart.
Use the bottom of a clean glass to press the center of each cookie and slightly flatten, leaving a slightly crinkled edge on each cookie. This is for filling with the key lime frosting later.
Bake for 13-15 minutes or until the edges of the cookies are set.
Remove the cookies and let them cool completely.
While the cookies cool, make the key lime frosting.