Preheat oven to 350°F. Line 2 baking sheets with parchment paper.
In a large bowl, beat butter, brown sugar, and granulated sugar until light and creamy, about 2 minutes.
Beat in the egg and vanilla until smooth.
In a separate bowl, whisk flour, cinnamon, nutmeg, baking soda, and salt.
Stir dry ingredients into the butter mixture just until combined.
Fold in oats, diced apple, raisins, and walnuts until evenly distributed.
Scoop dough into 1.5 tablespoon portions and place 2 inches apart on prepared sheets.
Bake 10 to 12 minutes, until edges are set and centers look just slightly soft.
Cool on the sheet for 5 minutes, then transfer to a rack to cool completely.
Notes
For thicker, chewier cookies, chill the dough for 20 minutes before scooping. Store in an airtight container for up to 4 days or freeze up to 2 months.