Lightly grease a loaf pan and line the bottom with parchment paper
Put the grated carrots and ginger in a mixing bowl and set aside.
Put the flour, baking powder, baking soda, spices, and salt in a mixing bowl and whisk together until well combined. Add the flour mixture to the carrots and mix well.
In a separate mixing bowl, combine both sugars, eggs, yogurt, and vanilla extract. Whisk everything together and slowly drizzle in the vegetable or coconut oil. Keep whisking until smooth.
Pour the flour mixture into the egg mixture 1/3 at a time and gently stir everything until it is combined. Pour into your prepared cake pan and bake at 350 degrees for 35 minutes. Lower the heat to 325 degrees, and bake for 20-30 more minutes until a toothpick comes cleanly out of the center.
To make the frosting, whisk together Greek yogurt, Confectioner’s sugar, and vanilla extract until smooth. Taste and adjust the sugar level to your liking. Let it set in the refrigerator while the cake bakes.
When your cake is done baking, take it out of the oven and let it cool in the pan for 15 minutes. Remove the cake from the pan and let it cool completely before you frost it.