Preheat oven to 350 degrees F (175 degrees C). Grease an 8-inch square baking pan.
Sift matzo cake meal, cream of tartar, baking soda, and salt together in a large bowl.
Beat butter in a large bowl with an electric mixer until creamy. Add white sugar gradually, beating until butter is fluffy. Beat in mashed banana and egg. Add matzo meal mixture in 3 batches, alternating with milk, beating batter briefly after each addition.
Spread batter in the prepared cake pan. Sprinkle brown sugar and cinnamon on top.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 30 minutes.