Gather all of the ingredients and preheat the oven to 350°F. Line a baking sheet with parchment paper.
Add the room-temperature salted butter, brown sugar, and granulated sugar to a medium mixing bowl and cream with a hand mixer on medium speed until smooth and slightly lighter in color.
Add the egg and 1 teaspoon vanilla extract, then mix on medium speed until fully incorporated. Scrape the bottom and sides of the bowl as needed.
Place the graham crackers in a food processor and pulse until they resemble fine sand.
Sift the processed graham crackers through a fine-mesh strainer into a medium mixing bowl, returning any coarse pieces to the food processor if necessary.
Whisk the graham cracker flour, all-purpose flour, baking powder, baking soda, and cinnamon together until evenly combined.
Add the dry mixture gradually to the creamed butter mixture, mixing only until a cohesive dough forms and no dry flour remains.
Place the dough on a sheet of parchment paper, cover it with a second sheet, and roll it evenly to about 1/4 inch thick.
Cut the dough into small shapes with miniature cookie cutters. Gather and reroll the scraps until all of the dough has been used.
Arrange the cookies on the prepared baking sheet with a little space between each piece.
Bake for 8 to 10 minutes, checking at the 8-minute mark, until the centers look dry and the edges are lightly golden.
Let the cookies rest on the baking sheet for a few minutes, then transfer them to a wire rack to cool completely and become crisp.
For the frosting, combine the room-temperature unsalted butter, powdered sugar, remaining 1 teaspoon vanilla extract, and whole milk in a medium mixing bowl.
Mix the frosting on low speed until the powdered sugar begins to incorporate, then increase to medium speed and beat until smooth and creamy.
Adjust the frosting with milk by the teaspoon if it is too thick or with additional powdered sugar if it is too loose.
Fold in the rainbow sprinkles with a spatula, then serve the frosting with the cooled miniature cookies.