Brown the butter in a small skillet on medium heat for 5-7 minutes, stirring occasionally. Remove from heat and cool for 10 minutes.
Using a stand mixer, mix bananas until smooth.
Add browned butter and sugars, mix on medium speed for 2 minutes.
Add eggs and vanilla extract, mix for 2 minutes.
Add greek yogurt and milk, mix until combined.
Add dry ingredients and mix until all dry bits are completely mixed in.
Place batter in a lined 8x8 baking dish and bake at 350F for 45-48 minutes.
While cake bakes, make the caramel sauce by adding butter and brown sugar to a saucepan on medium heat, let melt for 5 minutes, slowly add heavy cream until thick caramel forms (about 2 minutes), remove from heat.
For frosting, add butter and cream cheese to stand mixer with whisk attachment, mix on high for 2 minutes.
Add confectioners sugar, vanilla, and cinnamon to frosting, mix on high for 2 minutes.
Once cake completely cools, poke holes in it and add caramel, allow to seep in, then frost and enjoy.
