In a mixing bowl, combine the cake mix, eggs, and oil. Stir until well combined. Chill the dough while the oven preheats, 15-20 minutes.
Preheat the oven to 350 degrees. Line a sheet pan with parchment paper.
Using a 1-tablespoon cookie scoop, portion the dough out onto the sheet pan. Bake for 8-10 minutes. The cookies will be set but will look slightly underdone in the middle. Let the cookies cool on the pan for 5 minutes before transferring to a cooling rack.
For the frosting, add the butter to a small skillet set over low heat. Melt the butter and continue to heat until the butter turns golden brown, stirring occasionally. Brown bits will form in the bottom of the pan. This will take about 5 minutes. Cool slightly.
In a mixing bowl, combine the powdered sugar, maple extract, brown butter, and cream. Stir or whisk until the frosting has a smooth consistency. Additional cream can be added a little at a time to achieve the desired spreading consistency.
Using an offset spatula, top each cookie with the frosting.
Store the cookies in an airtight container with parchment paper between the layers. The cookies will keep for up to 4 days.