Preheat the oven to 350°F (175°C). Grease a 9 x 5-inch (23 x 13 cm) metal loaf pan with the softened butter. Line the bottom and long sides with parchment paper, leaving enough overhang to lift out the baked loaf.

Place the 10 tablespoons (142 g) butter in a light-colored saucepan over medium heat. Cook for 5 to 8 minutes, swirling or stirring frequently as it melts, foams, and begins to brown.

Continue cooking until the foam subsides slightly and amber-brown milk solids appear on the bottom of the pan. The butter should smell nutty rather than burned. Remove it from the heat immediately and pour it, including the browned solids, into a heatproof bowl.

Let the butter cool for about 15 minutes. It should remain liquid but no longer feel hot enough to cook the eggs when mixed.

In a medium bowl, whisk together the flour, baking soda, baking powder, cinnamon, and salt until evenly distributed.

In a large bowl, whisk the mashed banana and brown sugar until the sugar is moistened and no firm banana pieces remain.

Whisk the eggs, sour cream, and vanilla into the banana mixture. Slowly pour in the cooled brown butter while whisking. The mixture should look smooth and glossy, with small banana fibers still visible.

Add the dry ingredients to the wet mixture: Fold gently with a flexible spatula until only a few dry flour streaks remain.

Fold in 1/2 cup (55 g) of the toasted walnuts, if using. Stop mixing as soon as the batter is evenly combined. It should be thick and spoonable rather than runny.

Scrape the batter into the prepared pan and smooth the surface. Sprinkle the remaining 1/4 cup (30 g) toasted walnuts over the top, if using.

Bake on the center rack for 55 to 65 minutes. Begin checking at 50 minutes. If the top becomes deeply browned before the center is done, cover it loosely with aluminum foil.

Insert a wooden skewer into the center: It should come out with a few moist crumbs but no wet batter. An instant-read thermometer inserted into the middle should register approximately 200°F to 205°F (93°C to 96°C).

Place the pan on a wire rack and let the bread cool for 15 minutes. The loaf should begin pulling slightly away from the sides but still feel delicate.

Use the parchment overhang to lift the loaf from the pan. Place it directly on the wire rack and cool for at least 1 hour before slicing. Cutting it while hot can compress the tender center and make it appear underbaked.