Preheat oven to 180ºC (350ºF). Line a square 9-inch x 9-inch (22cm x 22cm) baking tin with parchment paper and set aside.
Melt butter and set aside for 5 minutes to cool down or until it's just warm, not hot.
Prepare a large mixing bowl, using either electric hand beaters or a stand mixer with the paddle attached.
In the mixing bowl, pour in cooled melted butter, eggs, sugar and vanilla. Beat on medium speed for 2 minutes until well combined.
With the beater running on low-medium speed, slowly drizzle in the olive oil in a steady stream to emulsify, about 1 minute in total.
Sift over flour and cocoa powder. Beat on low-medium speed until incorporated, ensuring not to over-beat.
Pour batter into the lined tin and spread out evenly.
Bake for 18-20 minutes or until a toothpick inserted into the middle comes out with some moist crumbs, not completely clean. Slightly under-baking will create a fudgy texture.
Remove from oven and allow to cool in tin for 10 minutes.
Carefully pull brownie out from the tin and transfer to a wire rack, leaving the parchment paper intact as it will still be soft and delicate to handle.
Allow the brownie to cool completely before starting the frosting, this may take a couple hours.