Basically I was trying to emulate a carrot cake with the rum, carrot, and spices as the cake itself, with egg white on top as a sort of "frosting."
I've also thought about infusing the maple syrup with the cinnamon and cloves. If you're looking for ratios, I'd say about a tablespoon of cloves and 2-3 cinnamon sticks per cup, depending on your personal preference.
1 oz. spiced rum (Sailor Jerry's)
1 oz. carrot juice
.3 oz maple syrup
1 dash orange bitters
Cinnamon stick, broken up (I only use about 1/3 of one)
.75 oz. egg white
Dry shake all ingredients for about 30 seconds until you achieve a nice layer of egg white foam on top of your drink. Add ice and shake again, then double strain into a coupe glass. Garnish with a sage leaf or thyme sprig.