Preheat oven to 350ºF. Grease a mini-muffin tin with olive oil or cooking spray. (If you don't have a mini-muffin tin, a sheetpan will work too.)
Place the carrots in the food processor and process them until they're coarsely chopped.
Add the walnuts, cheese, garlic, salt and pepper to the food processor and pulse until chopped. Add the egg and pulse again to incorporate.
Form the carrot mixture into 16 balls, squeezing them a little to make sure they hold their shape, and place each one in a cup in the muffin tin.
Bake the meatballs for about 45 minutes, turning halfway through cooking time, until they're golden-brown and slightly crispy on the outside.
Remove the tin from oven and cool for 5 minutes; to remove the meatballs, run a butter knife along the edges of each muffin cup to loosen before gently popping them out.