Preheat the oven to 350 °F (175 °C)
In a medium saucepan, melt the butter over medium heat until the butter is deep brown (about 3 minutes)
Remove the butter from the heat and whisk in the cardamom.
In a mixing bowl, mash the plantains using a potato masher. Let the butter cool for a bit and then add it to the mashed plantains.
Add the yogurt, brown sugar, egg, and vanilla extract to the plantains. Mix the ingredients together using an electric mixture until you get rid of most of the lumps in the mixture.
In a separate mixing bowl, combine the flour, baking powder, baking soda, and salt.
Add the flour mixture to the plantain mixture. Use a spoon to fold the flour mixture into the plantain mixture until no dry flour remains. Do not over-mix, the mixtures should be just combined.
Spray a 9-by-5-inch loaf pan with cooking spray. Pour the batter into the loaf pan, making sure that it spreads evenly around the pan.