In a mixer bowl fitted with paddle attachment, or using a hand-held mixer, beat together eggs, granulated sugar, and brown sugar on medium-high speed until light and fluffy, about 3 minutes. On low speed and with the mixer running, add oil slowly and beat until combined. Beat in vanilla extract until combined, then turn off mixer. Fold in carrots by hand until combined. Fold in flour mixture just until combined. Don’t overmix. Fold in nuts and shredded coconut.