In a large bowl, using an electric mixer, cream the ¾ cup unsalted butter and 1 cup granulated sugar and ½ cup brown sugar until light and fluffy, about 2 minutes. Add the 1 large egg and mix thoroughly.
In a medium bowl, whisk together the 1 ¾ cups oat flour, 1 teaspoon baking powder, ¼ teaspoon baking soda, ½ teaspoon ground cinnamon, ½ teaspoon ground gingerr, and ½ teaspoon fine sea salt, then mix it into the butter mixture until well combined. Add the 2 cups rolled oats and 1 ½ cups chopped walnuts on low speed and mix until well blended.
Place the dough in the fridge to chill for 1 hour or up to 3 days.
When ready to bake the cookies, heat the oven to 350°F and line two baking sheets with parchment paper.
Scoop out tablespoons of batter, roll them into balls, and place them on the prepared sheets about 1 inch apart. Place in the oven and bake for 15-20 minutes until lightly browned. Remove cookies from oven, cool on pan for 5 minutes, then place cookies on a wire rack to cool completely.
Cookies may be stored in an airtight container at room temperature for 5 days.
Notes
Oat substitutes. If you can't eat oats, swap brown rice flour for the oat flour. Instead of oats, try quinoa flakes or flaked coconut.
Brown the butter. For depth of flavor, brown the butter before adding it to the dough. If you choose to take this step, make sure to cool the butter slightly before proceeding with the recipe (you don't want it to be too hot to cook the egg!). You could use a whisk and spoon rather than a mixer to streamline the process if you go this route.
Dairy-free options. Substitute non-dairy butter for the butter or use coconut oil (refined or unrefined coconut oil is fine). You could melt the coconut oil first or keep it semi-solid and cream it with the sugars.
Egg-free. I haven't made these cookies egg-free, but you could try 1 flax egg in place of the egg to hold the dough together, and it will probably work great.
Refined sugar-free. Use all coconut sugar in place of the granulated and brown sugars.