Preheat oven to 425°F. Spray a baking sheet with cooking spray.
Grate 1 carrot and cut the remaining into sticks, set aside separately.
Whisk together the flours, brown sugar, baking powder, and salt. Slice the cold butter into smaller pieces, then cut into the flour using 2 knives, a fork, or a pastry blender; mixture should be well combined and crumbly.
Stir in the yogurt, 5 teaspoons milk, and vanilla. Fold in the grated carrots.
Divide the dough into 8 balls and place each onto the prepared baking sheet. Slightly flatten the balls with the bottom of a drinking glass. Brush the remaining teaspoon of milk over the tops of the scones and then sprinkle with sugar.
Bake for 12-15 minutes until lightly golden. Allow to cool for 5 minutes, then remove from pan and place onto a wire rack to finish cooling.
Serve scones with sliced strawberries and carrot sticks on the side.