Preheat the oven to 350°F (180°C). Line 2 12-cup muffin tins with paper liners and grease with butter spray.
Make the base carrot cupcake batter: In a food processor, process the carrots until finely chopped.
In a large bowl, whip the butter with an electric hand mixer on medium speed until fluffy, about 2 minutes. Add the granulated and brown sugars, and beat until smooth, about 2 minutes, scraping down the sides of the bowl as needed.
In a small bowl, beat together the eggs and vanilla extract.
Add the egg mixture to the butter mixture in thirds, beating on medium speed between each addition until fully incorporated.
In a medium bowl, combine the flour, baking powder, baking soda, salt, cinnamon, ginger, and nutmeg.
Add the dry ingredients to the wet ingredients in thirds, beating on low speed between each addition until incorporated.
Add the coconut oil and beat on medium-low speed for 1 minute, until fluffy.
Finish making the cupcakes according to your desired variation.
For the Cheesecake-Stuffed Carrot Cupcakes: Gently fold the carrots and 1⅓ cups Fisher Walnuts into the batter.
Make the cheesecake filling: In a large bowl, whip together the cream cheese and sour cream with an electric hand mixer on medium-low speed until fluffy, 2 minutes. Add the granulated sugar and vanilla extract, and beat until fully incorporated. Add the eggs and beat on low speed until smooth.
Spoon the carrot cupcake batter into the prepared muffin cups, filling each a little more than halfway. Tap the muffin tins on the counter to smooth the batter. Spoon the cheesecake filling over the batter, filling each cup ¾ of the way. Bake the cupcakes for 25–27 minutes, or until a toothpick inserted in the center comes out clean. Remove the cupcakes from the oven and let cool completely.
Meanwhile, make the lemon vanilla cream cheese frosting: In a large bowl, whip together the cream cheese and vanilla extract with an electric hand mixer on medium speed until fluffy, 2 minutes, scraping down the sides of the bowl as needed. Add the powdered sugar and beat on low speed until smooth. Add the lemon zest, lemon juice, and salt, and mix on low speed to incorporate. Transfer the frosting to a piping bag fitted with a large round tip and refrigerate until ready to use.
To frost, place the tip of the piping bag upright at the center of a cupcake and squeeze until the cupcake is covered with frosting. Repeat with the remaining cupcakes, then garnish the centers with the remaining cup of Fisher Walnuts.