Preheat the oven to 350°F (175°C). Line an 8×8 inch baking pan with parchment paper and grease lightly.
In a large bowl, cream together the softened butter and granulated sugar until light and fluffy.
Add the egg and vanilla extract to the butter mixture and beat until combined.
In a separate bowl, whisk together the all-purpose flour, baking powder, baking soda, and salt.
Alternately add the dry ingredients and buttermilk to the wet mixture, beginning and ending with the dry ingredients. Mix until just combined.
Fold in the mashed bananas gently until evenly distributed.
Pour the batter into the prepared baking pan and spread evenly.
Bake for 22-25 minutes, or until a toothpick inserted into the center comes out clean.
Allow the cake to cool completely in the pan on a wire rack.
Once cooled, crumble the cake into a large bowl until fine crumbs form.
Melt 6 ounces of the white chocolate chips in a microwave-safe bowl in 30-second intervals, stirring between each, until smooth.
Add about 2 tablespoons of melted chocolate to the cake crumbs and mix thoroughly until the mixture holds together when pressed.
Roll the mixture into 24 evenly sized balls and place them on a parchment-lined baking sheet.
Insert a lollipop stick into each cake ball and refrigerate for at least 1 hour to firm up.
Melt the remaining 6 ounces of white chocolate chips in the microwave in 30-second intervals, stirring until smooth.
Dip each cake pop into the melted chocolate, allowing excess to drip off, then place back on the parchment paper to set.
Refrigerate the cake pops until the chocolate coating is firm, about 30 minutes, before serving.