In a large bowl, whisk together the oil and sugars.
1/3 cup (80ml) vegetable oil or canola oil, 1/2 cup (100g) lightly packed brown sugar, 1/2 cup (100g) granulated sugar
Add sour cream, vanilla and eggs until smooth.
1/2 cup sour cream, 1 tsp vanilla extract, 2 large eggs
Add the dry ingredients to the wet ingredients and stir until just combined, then fold in the shredded carrots. Spoon the batter into a muffin pan lined with cupcake liners.
1 cup finely grated carrot
Bake for 18-20 minutes or until a toothpick inserted into the center comes out clean. Cool the cupcakes in the pan for 10 minutes then remove from pan and transfer them to a wire rack to cool completely.
Frosting:
In a large mixing bowl, beat the cream cheese, butter, and vanilla together until combined.
4 oz (1/2 block) full-fat cream cheese, 1/2 cup (1 stick) unsalted butter, 1/2 tsp vanilla extract
Mix in the powdered sugar until smooth. Chill in the fridge for 20 minutes or so if needed to thicken.
2 cups (240g) powdered sugar
Top each completely cooled cupcake with a dollop of frosting and spread it out with an offset spatula or the back of a spoon. Sprinkle with some chopped walnuts/pecans or a light dusting of cinnamon if desired.
1/4 cup finely chopped walnuts/pecans, 1 tsp ground cinnamon
Notes
Leftovers can be stored in an airtight container in the fridge for 3-4 days. Bring to room temperature before serving.