Preheat oven to 350°F. Grease a 9x13-inch baking pan with cooking spray and line with parchment paper.
Combine flour, cocoa powder, baking powder, baking soda, and salt in a large bowl.
Combine the brown sugar, granulated sugar, buttermilk, and oil in a separate large bowl or in the bowl of a stand mixer. Beat on medium speed until the mixture is smooth and combined.
Beat in eggs, one at a time, until combined. Beat in vanilla.
Combine the instant coffee and water in a liquid measuring cup. With the mixer running on low speed, alternately add the dry ingredients and coffee mixture in 3 additions, starting and ending with the dry ingredients.
On the last addition of the dry ingredients, stop mixing with the mixer and switch to a rubber spatula or wooden spoon to finish mixing until no streaks of flour remain.
Pour the cake batter into your prepared pan and bake 25 to 30 minutes or until a toothpick inserted into the center of the cake comes out clean.
Allow the cake to cool completely in the pan before removing to a wire rack.