Preheat oven to 350°F (180°C) and place the oven rack in the middle of the oven. Butter or spray a 12 - 1/2 cup muffin pan.
In a large bowl combine the flour, sugars, baking powder, baking soda, salt, and white chocolate. Set aside.
In a medium-sized bowl combine the mashed bananas, eggs, melted butter, and vanilla. With a rubber spatula or wooden spoon, lightly fold the wet ingredients (banana mixture) into the dry ingredients until just combined and batter is thick and chunky. (The important thing is not to over mix the batter. You do not want it smooth. Over mixing the batter will yield tough, rubbery muffins.)
Spoon the batter into the prepared muffin tins and place a slice of banana on top of each muffin for garnish.
Bake about 20 - 25 minutes or until a toothpick inserted in the center comes out clean. Place on a wire rack to cool for five minutes and then remove muffins from pan.
Serve warm or at room temperature.