Place rack in center of oven and preheat oven to 350 degrees F.
Whisk together flours, baking soda, cinnamon, nutmeg and salt in a medium-sized bowl. Combine butter, oil and sugars in the bowl of a stand mixer and mix on high speed, scraping down sides if necessary, until sugars have dissolved and mixture is light in color, about 1 minute. Add egg, carrot puree and vanilla and beat an additional 30 seconds. Add flour mixture and beat an additional 30 seconds. Add oats, flakes, raisins and walnuts and mix over low speed just until incorporated. Dough will be slightly sticky and less cohesive than traditional cookie dough. Line a large cookie sheet with parchment paper. Using between 3 to 4 tablespoons of batter, form a ball and place on cookie sheet. Repeat with remaining batter, leaving about 3 inches between cookies. Wet hands and use palm of hand to flatten cookies until about 1/4-inch thick. Bake for 12 minutes, until cookies are fragrant but still soft. Let cookies cool slightly, then transfer to a wire rack to cool completely.
Per cookie (12): Calories: 189; Total Fat: 9.5 grams; Saturated Fat: 2 grams; Protein: 3 grams; Total carbohydrates: 24 grams; Sugar: 11 grams; Fiber: 2 grams; Cholesterol: 21 milligrams; Sodium: 125 milligrams