Keep the unsalted butter near 68 to 70°F before mixing. Beat it with the light brown sugar and granulated cane sugar for 2 to 3 minutes. Use medium speed and scrape the bowl once at minute 1. Add the large egg and both extracts. Then mix for 30 seconds until the streaks disappear. The mixture should turn pale tan and look fluffy with faint sugar crystals. You'll smell vanilla and banana before the surface becomes slightly glossy. The texture should stay soft. Stop before minute 4 because excess air can make the cookies spread unevenly. Cream only 3 minutes because controlled aeration keeps the centers thick.

Keep the creamed mixture near 68 to 70°F. Whisk the all-purpose flour and banana cream instant pudding mix with baking soda and fine sea salt for 30 seconds. Add that blend to the bowl. Fold with a rubber spatula in a J-shaped motion for 12 to 15 strokes. Then add the white chocolate baking chips and crushed wafers with six gentle folds. The dough should look thick and pale beige with uneven chips and wafer pieces throughout. It should pull from the bowl in soft clumps without wet streaks. Stop once the last flour patch disappears. Overmixing past 20 strokes makes the baked centers tough. Fold briefly because limited gluten keeps the cookies tender.

Cover the dough and chill it at 38 to 40°F for 30 minutes. Scoop twelve portions weighing 42 to 45 grams each. Roll each portion for about 5 seconds and press it to a 3/4-inch thick mound. Arrange the portions on a lined 13×18-inch tray with about 2 1/2 inches between them. The chilled dough should feel firm but still yield under your fingertips. Each mound should hold rough edges instead of slumping flat. Press the reserved chips and wafer pieces unevenly across the tops. (I skipped chilling once, the cookies merged at the edges.) Shape while cold because firm butter delays spreading in the oven.

Bake the twelve chilled portions in a 350°F preheated oven for 11 to 13 minutes. Place the tray on the center rack. Rotate it front to back at minute 7 with one smooth turn. Remove the tray when the edges turn golden and the centers still look slightly matte and raised. The wafer pieces should smell toasted and the white chocolate chips should look soft but not runny. Small cracks will form across the pale golden tops. Go past minute 14 and the centers lose their chewy texture. Bake at 350°F because moderate heat sets the edges before drying the middle.

Leave the cookies on the warm tray at about 70°F for 10 minutes. Then transfer them and rest for another 15 minutes before serving. Lift each cookie from underneath so the soft center stays intact. Break one cookie into two uneven halves and place it on a small white dessert plate. The interior should look soft and dense with melted white chocolate pockets and crumbly wafer pieces. The edges should stay lightly crisp while the center bends without looking raw. Move them too soon and the warm cookies can split. Cool for 25 minutes because carryover heat finishes setting the starch.