In a stand-up mixer with the paddle attachment, beat the butter and sugar until the butter is broken up and homogenous with the sugar. Pour in the oil and beat on medium speed for about 2 minutes.
Add the eggs one at a time, allowing the first to fully incorporate before adding in the second.
Measure out the flour and remove about 2 – 3 tablespoons and set aside. Mix the remaining dry ingredients, except the carrot and walnuts.
Toss the carrots and walnuts in a separate bowl with the 2 – 3 tablespoons of reserve flour until it’s all combined—this will help keep the carrots and nuts from sinking to the bottom of the cupcakes.
Add the dry ingredients all at once and beat for just a few seconds on low until it’s about 50% combined. Add the carrot-walnut mix and beat a few seconds more, until well combined.
Scrape down the sides and bottom of the mixing bowl with a rubber spatula to make sure it is all incorporated and evenly mixed.
Line two muffin tins with paper cupcake wrappers and divide the batter evenly among them (a slightly heaped 2-ounce, or roughly 2-1/4” diameter, ice cream scoop does the job perfectly).
Bake for 22 – 25 minutes.
Allow them to cool for about 10 minutes in the muffin tins before removing the cupcakes and setting on a wire cooling wrack until completely cooled.
To make the frosting, whip the cream cheese and butter in a stand-up mixer with the whisk attachment just so they’re a little combined. Add in the powdered sugar and vanilla, and beat slowly at first (to avoid a powdered sugar cloud filling the room) before increasing the speed and whipping until it’s perfectly combined. If you want something a little lighter, and not so heavy and dense, add a little heavy cream and whip for about a 2 – 3 minutes, or until it’s a little fluffier. Frost them either by hand or piping bag.
If you’re unsure of everyone’s reaction to Five Spice, or introducing people to the spice, I would go with 2 t-spoons; those familiar and enthusiastic with it can up the amount to 3 t-spoons/1 tablespoon to get a stronger taste without it being overpowering.
The cupcakes and frosting really are best at room temperature. Because the cupcakes are so moist, though, you can make and frost them in advance, stash them in the fridge, and just let them come to room temperature before serving.