Cut the butter into smaller chunks and place in a light-colored pan over medium heat. (See recipe notes about browning the butter for the banana bread and the frosting at the same time to save you from doing it twice).
As the butter melts, it will start to bubble and make a crackling sound. Keep going, stirring occasionally.
After the crackling subsides, the butter will start to brown quickly, so keep an eye on it. A foam may form on top, you can just use your spoon to push it to the side to check the color of the butter.
When the butter is a deep golden brown color, and you can smell an irresistible nutty aroma, the brown butter is ready! Remove from the heat and pour into a heatproof bowl immediately to prevent any further browning.
For more info on browning butter, check out my blog post - How to Make Brown Butter.
If you are browning just the butter for the banana bread (1 stick of butter), continue with the banana bread recipe. If you are browning butter for the banana bread and the frosting at the same time (2 sticks), make sure you split your brown butter in half before continuing with the recipe (see recipe notes below).
Make the Banana Bread
Grease and line a 9x5 loaf pan and preheat your oven to 350°F / 180°C.
Put your liquid brown butter in a large bowl along with the sugar and light brown sugar. (If you made your brown butter earlier, be sure to melt it in the microwave to bring it back to a liquid state). Mix everything together with a whisk.
Peel your bananas and mash them in a small bowl with a fork or potato masher. Add them to the butter/sugar mixture and mix with the whisk.
Add the eggs, vanilla, yogurt, and vegetable oil and whisk everything in until combined.
Sift in the dry ingredients, including the flour, baking powder, baking soda, and salt.
Chop the pecans roughly and mix them into the batter - you may want to switch to a spoon or spatula for this part, as the pecans can get stuck in the whisk.
Pour the batter into your prepared loaf pan and bake for 55 minutes, or until a skewer inserted into the middle of the banana bread comes out clean.
Make the Brown Butter Frosting
The reserved brown butter you prepared earlier should have solidified by now. If you had it in the fridge, be sure to bring it to room temperature before using. Give it a good mix to evenly disperse all the milk solids (little brown specks). Take 3 tablespoons and add them to the bowl of a stand mixer (or regular bowl with an electric whisk).
Add the cream cheese (also at room temperature) and beat together with the brown butter, just until combined.
Add the powdered sugar and mix in slowly to avoid a sugar explosion all over your worktop. Once it's all incorporated, add the vanilla and a pinch of salt, then turn the mixer to high speed for about a minute.
Spread over the top of your cooled banana bread and enjoy.
Notes
Browning the butter - Both the banana bread and the frosting use browned butter. I recommend browning the butter for both at the same time, to avoid having to do the process twice. This also works well as the banana bread calls for 'melted butter' and the frosting calls for room temperature butter, so it gives the brown butter time to resolidify before you make the frosting. There are a few things to keep in mind:
I recommend browning 2 sticks of butter, this is actually a little more than you'll need, but it will be easier to divide the browned butter into two, and you'll have some lovely brown butter leftover to spread on your banana bread.
When dividing the brown butter, you can do it by eye, but the easiest and most accurate way to do it is with a kitchen scale so you can make sure you have equal amounts.
Try to make sure you have an even amount of milk solids (the little brown bits) in each lot of the brown butter, use a spoon to scoop some out and transfer between the two if you need to adjust.
When the half batch of the frosting solidifies, be sure to give it a good mix to disperse the milk solids well throughout the brown butter.
Alternatively, you can prepare all the brown butter ahead of time and let it all solidify, then give it a stir and split it in half when you are ready to bake - melting half of the brown butter for the banana bread by microwaving in 30-second increments.