Preheat the oven to 350°F/175°C fan. Line a narrow, tall loaf pan with parchment paper.
Place the mashed bananas in a large bowl, then whisk in eggs, sugar, yogurt, olive oil, and vanilla until smooth.
Toss the blueberries with 2 tablespoons of flour. Fold the flour, baking soda, baking powder, and white chocolate into the banana mixture, then gently fold in the blueberries.
Pour the batter into the prepared pan and arrange the banana halves on top. Bake for 40–45 minutes until golden and a skewer inserted comes out clean. Cool completely before slicing.