Preheat the oven to 350°F (175°C). Grease a Bundt pan well and set aside.
In a medium bowl, whisk together the flour, baking powder, and salt.
In a large bowl using a hand mixer, or in the bowl of a stand mixer, beat the softened butter until light and fluffy.
Add the granulated sugar and continue beating until well combined.
Mix in the vanilla extract, then add the eggs one at a time, beating well after each addition.
Mash the bananas and stir them into the batter.
Add half of the dry ingredients and mix until just combined. Add the Greek yogurt and mix again, then fold in the remaining dry ingredients until no streaks of flour remain. Be careful not to overmix.
Pour the batter into the prepared Bundt pan and smooth the top.
Bake for 55 to 65 minutes, or until a toothpick inserted into the centre comes out clean.
Let the cake cool in the pan for 15 minutes before carefully inverting it onto a wire rack to cool completely.
While the cake cools, prepare the icing. In a small saucepan over medium heat, melt the butter with the brown sugar and milk. Stir until smooth, then remove from the heat and mix in the vanilla. Whisk in the powdered sugar until smooth and slightly thickened.
Drizzle the icing over the cooled Bundt cake and finish with chopped walnuts before serving.