Preheat oven to 350 degrees F. Line a large baking sheet with parchment paper or a Slipat baking mat and set aside.
Place the butter in a medium saucepan and melt over medium heat, stirring often, until it foams, smells nutty, and browns. This will take about 5 to 7minutes. Scrape the browned butter into a large bowl. Let cool to room temperature. You can put it in the fridge to speed up the process.
In a small bowl, whisk together the flour, cinnamon, baking soda, salt, and nutmeg, if using. Set aside.
Add the sugars to the brown butter and stir with a spatula until combined. Add the egg and vanilla extract and stir until smooth. Stir in the shredded zucchini.
Add the dry ingredients and mix until just combined. Stir in the oats and chocolate chips; don’t over mix.
Use a cookie scoop to scoop the dough (1.5 to 2 tablespoons) onto the prepared baking sheet, leaving about 2-inches in between the cookies.
Bake for 12 to 14 minutes or until lightly browned on the sides, but still soft in the centers.
Remove from the oven and sprinkle with flaky sea salt. Let the cookies cool on the baking sheet for 5 minutes. Transfer to a cooling rack and cool completely.
Store the cookies in an airtight container on the count for up to 4 days. You can also freeze the cookies for up to 2 months. They are really good frozen.