Preheat oven to 350 degrees F. Spray a 9x13 cake pan with baking spray & set aside. In a medium bowl, whisk together flour, baking powder, baking soda, salt, and chai. Set aside.
In a stand mixer or with a hand mixer, cream butter & sugars for 2-3 minutes until pale and fluffy. Beat in eggs and vanilla until well blended. Mix in mashed banana until combined, then fold in zucchini. Add flour mixture alternately with buttermilk in 3 batches and mix until well combined.
Pour batter into prepared pan and bake in preheated oven for 40-50 minutes, until golden brown and a toothpick inserted comes out clean. Remove from oven and cool in pan for at least an hour.
Brown Butter Cream Cheese
Set a heavy-bottomed pan over medium-low heat. Add the butter and allow it to melt. Gently swirl the butter as it goes through two stages - popping & crackling as the water boils out, and the browning of the butter solids. As the solids begin to brown, use a spoon to peek through the foam for the light brown flecks inside - it will start to smell nutty and delicious. The whole process from solid stick to brown butter will take about 10 minutes. Continue to swirl and stir, then remove it from the heat and pour it into a shallow bowl, allowing it to cool for about 20 minutes before transferring it to the refrigerator. Chill the brown butter until it's solid again, but still soft, about 25 minutes.
When the butter is solid again, transfer it to a medium mixing bowl. Use a mixer to whip the butter for 2 minutes until fluffy. Add the cream cheese and vanilla and whip about 3 minutes until well blended and fluffy.
Sift in powdered sugar, chai, and salt and blend until smooth. Add additional powdered sugar, vanilla, and/or chai to taste.
Spread the icing over the top of the cooled cake, and garnish with pecans as desired.