Preheat the oven to 350°F. Line a baking sheet with parchment paper. Set aside.
In a large bowl, or the bowl of an electric mixer fitted with the paddle attachment, beat the eggs and brown sugar on medium speed until combined. Add the oil, mashed bananas and mix until combined.
Add the flour, baking powder, baking soda, cinnamon, salt, and ground cloves. Mix until combined.
Use a 2 T sized scoop and portion the dough onto the baking sheet, leaving 2 inches between each cookie. Bake for 11-13 minutes, the tops should be puffy and should spring back when touched.
Remove from the oven and let cool on the pan for 5 minutes, then transfer to a wire rack to cool completely.
Make the frosting: Place the powdered sugar in a large bowl. Brown the butter in a small saucepan over medium heat, stirring occasionally until amber in color, and has a nutty aroma. Pour the brown butter over the powdered sugar. Add the vanilla, and whisk to combine. Add the milk, a tablespoon at a time until it reaches your desired consistency. Drizzle the frosting over each cooled cookie.