Peel the carrots and shred them using a bow grater or food processor until you have 1/2 cup of carrots. Lay the shredded carrots on a paper towel and press down with another paper towel to remove any excess moisture. Set aside.
In a medium mixing bowl, whisk together the beaten egg, milk, and butter. Set aside.
In a large mixing bowl, stir together the flour, baking powder, salt, pumpkin pie spice, and brown sugar.
Add the shredded carrots and walnuts to the dry ingredients, and then pour in the wet ingredients. Stir until just combined.
Let the pancake batter rest for 30 minutes. (If you need to cook pancakes right away, you can. The batter gets better if you have to let it sit, though.)
Place a large skillet or griddle over medium-high heat. Melt some butter in the skillet and then spoon 1/4 cup of pancake batter into the skillet for each pancake. Do not overcrowd the pan.
Cook until bubbles form on top of each pancake. Flip the pancakes over and cook until the bottom is golden brown. Repeat with the remaining batter.
Serve warm with maple syrup and extra walnuts, if desired.
Notes
To toast the walnuts, place a skillet over medium heat. Add the walnuts and cook for about 5 minutes, stirring occasionally, until they are slightly browned and smell toasted. Transfer the nuts to a bowl, and once cool enough to handle, chop them up.