Preheat the oven to 350°F. Line an 8-x-8-inch pan with foil so that it extends up the sides. Line the bottom of the pan with a square parchment. (You could probably get away with only the foil or parchment, but I always err on the side of caution. Plus, it makes for easy cleanup.)

Melt the butter in a medium saucepan over low heat. Remove from the heat and stir in the brown sugar, egg, and vanilla. Stir in the flour, baking powder, and salt until just combined. Gently fold in the carrots, blueberries, and white chocolate.

Pour the batter into the prepared pan, spreading it evenly. Bake until they are set in the center and a toothpick comes out clean, about 30 minutes.

Cool, then cut into bars. Store leftovers in the refrigerator.