Preheat oven to 350°F and grease a 9×13-inch baking dish with nonstick spray.
In a large bowl, cream the butter and sugars with an electric mixer until light and fluffy. Add eggs, one at a time, beating well after each addition. Mix in vanilla.
Whisk together flour, baking soda, and salt and add the mixture to the butter mixture in thirds, alternating with the bananas and Greek yogurt, mixing well after each addition.
Bake until a toothpick inserted into the center comes out clean, 35-40 minutes. Let cool completely on a wire rack.
While cake cools, make the frosting:
In a medium bowl, beat butter with an electric mixer until smooth. Gradually add powdered sugar and beat until combined.
Dissolve the instant coffee in milk, then add to the butter mixture and beat on high speed until light and fluffy.