Heat your oven to 350°F (175°C). Grease a 9x5-inch loaf pan well, or line it with parchment. Set it aside while you work on the batter.
Melt the butter in a small saucepan over medium heat. Let it simmer and stir often until it turns golden, foamy, and fragrant with brown bits at the bottom. This takes about five minutes, so don't walk away. Pour it into a bowl right away so it stops cooking.
Peel the three ripe bananas and add them to a large bowl. Mash them with a fork until mostly smooth. A few small lumps are fine and give nice pockets of banana flavor. Don't use a blender, because it makes the loaf gummy.
Pour the warm browned butter over the bananas and stir well. Add the brown sugar, white sugar, egg, and vanilla. Whisk until everything looks glossy and combined. Make sure the butter is only warm, not hot, so it doesn't scramble the egg.
Sprinkle in the flour, baking soda, cinnamon, and nutmeg. Fold gently with a spatula until you no longer see dry flour. Stop mixing right there. Overmixing makes banana bread tough and chewy instead of soft.
Scrape the batter into your prepared pan and smooth the top. Bake until a toothpick in the center comes out clean or with a few moist crumbs. Start checking around 50 minutes. If the top browns too fast, tent it loosely with foil.
Let the loaf rest in the pan for about ten minutes. Then move it to a wire rack. Cool it for at least 20 more minutes before slicing, so the crumb sets and cuts cleanly. It's also lovely warm, if you can't wait.