For the cookies:

1.  Preheat your oven to 350°F and line a baking sheet with parchment paper or a silicone baking mat.
2. In a light-colored skillet over medium-low heat, add the butter and allow it to brown for about 8-10 minutes, stirring occasionally with a rubber spatula. You’ll know that it’s done when it starts to smell like toffee, and the milk solids turn a light brown color. Once that happens, take it off the heat and allow it to cool.
3. Set aside ¼ cup of the butter for the frosting.
4. Once cooled, add the remaining butter to the bowl of your stand mixer (fitted with the paddle attachment) along with the granulated sugar and light brown sugar, and beat on medium-low (speed 2 on a KitchenAid) for 1-2 minutes until the mixture is fluffy and looks like wet sand.
5. Add in the eggs, one at a time, waiting until each egg is incorporated before you add the next.
6. Add the vanilla bean paste and mix on low for another 10-15 seconds until it’s incorporated.
7. Sift in the flour, baking soda, cinnamon, ginger, nutmeg, and salt, and mix on medium-low until just combined. 
8. Add the carrots and turn the mixer back to low for about 15 seconds until they’re incorporated.
9. Use a 1 ½ tablespoon cookie scoop to scoop the cookies onto the lined baking sheet and bake for 11-13 minutes. The edges will be golden, and the tops will be dry but still soft-looking. You don’t want to over-bake these cookies because they’ll lose their chewy texture.
10. Let the cookies sit on the baking sheet for 2-3 minutes to set before moving them to a wire cooling rack to cool completely. The cookies must be completely cooled before you frost them, or the frosting will just slide off the sides.

For the frosting:

1. In a large bowl, or the bowl of your stand mixer, add the cream cheese, ¼ cup of the cooled browned butter, milk, vanilla bean paste, and salt. Beat until combined.
2. Add the powdered sugar and mix on low until fluffy.
3. For a more rustic look, spread the frosting onto the cookies with an offset spatula, or for a more polished look, use a piping bag to pipe the frosting onto the cookies.
4. Sprinkle with chopped pecans if desired and serve.