Preheat the oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, nutmeg, and ginger until well combined.
In a large bowl, beat the granulated sugar, brown sugar, and eggs together until smooth and pale.
Add the Greek yogurt, vegetable oil, and vanilla extract to the sugar and egg mixture. Mix until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined. Do not overmix.
Fold in the grated carrots, chopped walnuts, and raisins until evenly distributed.
Pour the batter into the prepared cake pan and smooth the top with a spatula.
Bake for 35 to 40 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes. Then transfer to a wire rack to cool completely.
While the cake cools, prepare the yogurt frosting by beating together the Greek yogurt, softened cream cheese, powdered sugar, and vanilla extract until smooth and creamy.
Once the cake is completely cool, spread the yogurt frosting evenly over the top.
Slice and serve. Store any leftovers covered in the refrigerator.