Preheat oven to 350°F. Grease and line a 9×13 inch baking pan with parchment paper.
Whisk together oat milk, yogurt, and lemon juice. Allow it to sit for at least 5 minutes.
In a mixing bowl, whisk flour, baking powder, baking soda, and salt. Set aside.
In a stand mixer or large mixing bowl with an electric hand mixer, beat butter on high for 1 minute until smooth. Add both sugars, beat on high for 2 more minutes until creamed. Scrape down the sides.
Beat in eggs and vanilla on medium until combined (about 30 seconds). Scrape sides, then beat in mashed banana until just combined.
With the mixer on low, add dry ingredients in three parts alternating with the oat milk mixture. Mix until just incorporated; do not overmix.
Pour batter into the lined baking dish and make sure it is even.
Bake for 40 - 50 minutes, until a toothpick comes out clean. If the top browns too quickly, loosely cover with aluminum foil.
Allow the cake to cool completely before frosting. You can refrigerate to speed up the process.
While the cake cools, make the frosting by combining all ingredients in a large mixing bowl. Beat with a handheld electric mixer on medium for 1 - 2 minutes until fluffy and smooth. Alternatively, use a stand mixer with the paddle attachment.
Frost the cake and enjoy!