In a large bowl, whisk together eggs, light brown sugar, granulated sugar, oil, yogurt, and vanilla extract until thoroughly mixed. Gently fold in shredded carrots and prepared flour mixture, taking care not to overmix. Transfer batter to prepared pan, and bake until an inserted toothpick near the center comes out clean, about 55 to 60 minutes. If the cake is browning too fast towards the end of the baking time, loosely tent it with aluminum foil.