Place a rack in lower third of oven; preheat to 350°. Lightly coat a 13x9" metal baking pan with nonstick vegetable oil spray. Line bottom of pan with parchment paper, then lightly coat parchment. Peel 2 very ripe bananas, scrubbed, and reserve peels. Place bananas, in a large bowl.

Melt ½ cup (1 stick) unsalted butter in a medium saucepan over medium heat, stirring often, until it foams, then browns, 5–8 minutes. Scrape brown butter into bowl with bananas and whisk to break up bananas and combine; reserve pan. Let banana butter cool slightly.

Whisk 2½ cups (313 g) all-purpose flour, 2 tsp. baking powder, 1½ tsp. Diamond Crystal or 1 tsp. Morton kosher salt, and ½ tsp. baking soda in a medium bowl to combine. Set dry ingredients aside.

Add 1½ cups (300 g) granulated sugar to banana mixture and whisk until smooth. Add 4 large eggs, room temperature, ¾ cup sour cream, room temperature, ½ cup vegetable oil, and 1 Tbsp. vanilla extract; whisk until smooth and emulsified. Add reserved dry ingredients and whisk just until no streaks remain (batter may be lumpy).

Scrape batter into prepared pan and spread to edges. Gently bang pan on counter 2–3 times to eliminate any large air bubbles. Bake cake until top is golden and springs back when gently pressed, 28–30 minutes. Transfer pan to a wire rack and let cake cool in pan (about 2 hours).

Trim and discard ends of reserved banana peels; chop peels into 1" pieces. Place in reserved pan and pour in 1½ cups whole milk. Set over high heat and bring to a simmer. Reduce heat to low and cook (milk should be steaming or just barely simmering around edges of pan) 10 minutes. Strain banana milk through a sieve into a large bowl; discard peels. Chill until cold, about 30 minutes.

Add one 3.4-oz. box instant vanilla pudding mix, ¼ cup sour cream, 2 tsp. vanilla extract, and ¼ tsp. Diamond Crystal or Morton kosher salt to cold banana milk and whisk until smooth. Let pudding mixture sit 5 minutes to thicken.

Meanwhile, whisk 1 cup heavy cream in a small bowl to medium-stiff peaks.

Add whipped cream to pudding mixture and fold in with whisk until no streaks remain.

Run a small offset spatula around sides of cake to loosen. Turn cake out onto rack and carefully peel away parchment; discard. Transfer cake to a platter. Scrape banana cream mixture on top of cake and spread out until nearly to edges, swooping and swirling decoratively. Chill cake at least 1 hour to further set banana cream if desired.

Just before serving, arrange 1 medium banana, peeled, sliced ¼" thick, and mini vanilla wafer cookies over banana cream as desired.