For cake:

1. Melt the butter over medium-low heat in a light-colored pot. Once melted, whisk constantly until the butter froths and then begins to turn brown and smell nutty. Keep whisking until it is studded with deep-caramel-brown flecks, then immediately take off the heat (butter goes from brown to burnt very rapidly, so keep close watch). Set aside, and allow to cool.
2. Preheat the oven to 350ºF, and line two cupcake pans with cupcake liners.
3. In a medium bowl, sift together the flour, sugar, baking soda, spices, and salt.
4. In the bowl of a stand mixer, beat together the eggs, vegetable oil, and cooled brown butter until well incorporated and no streaks of egg yolk remain.
5. Add the dry ingredients in three additions, scraping down the sides of the bowl with a rubber spatula between additions. Mix together until just combined.
6. Fold in the grated carrot and pecans, if using.
7. Divide the batter among the cupcake molds, filling each mold a scant 3/4 full.
8. Bake for 14-18 minutes, or until a toothpick or cake tester inserted into one of the cupcakes comes out clean.
9. Allow to rest for 5 minutes, then move to a cooling rack and allow to cool completely before frosting.

For frosting:

1. In the bowl of a stand mixer, beat together the butter and sugar using the paddle attachment, until lightened in color and fluffy and no streaks of butter remain.
2. Add the cream cheese, and mix together until the cream cheese is completely incorporated.
3. Frost the cooled cupcakes, and garnish with a whole pecan, if desired.