Brown the butter first so it has time to cool. In a light-colored saucepan over medium heat, melt the butter, stirring frequently. Continue cooking for 4 to 5 minutes. The butter will foam, then the foam will subside, and you will see small amber-brown bits forming on the bottom of the pan. When it smells nutty like toasted hazelnuts, pull it off the heat immediately and pour it into a large mixing bowl. A light-colored pan is important here so you can see the color change clearly.

Let the brown butter cool in the bowl for 10 full minutes. This step matters. If the butter is too hot when you add the sugars and egg, it will scramble the egg and your dough will be grainy.

While the butter cools, shred the zucchini on the fine side of your box grater. Wrap it in a clean kitchen towel and squeeze firmly over the sink until no more liquid comes out. Set the dry zucchini aside.

Preheat your oven to 350 degrees F. Line two baking sheets with parchment paper.

Whisk both sugars into the cooled brown butter until well combined. Add the egg and vanilla and whisk again until the mixture looks smooth and slightly glossy, about 1 minute.

Stir in the squeezed zucchini. Then add the flour, baking soda, baking powder, salt, and cinnamon all at once. Switch to a wooden spoon or spatula and stir until just combined. Fold in the chopped walnuts. The dough will be soft and slightly sticky.

Scoop rounded tablespoon-sized portions of dough onto the prepared baking sheets, spacing them 2 inches apart. If the dough feels too soft to scoop cleanly, refrigerate it for 10 minutes before scooping.

Bake for 11 to 13 minutes until the edges are visibly golden and the centers look just barely set when you peek at them. They will firm up as they cool. Pull them at 11 minutes if you want a softer center, 13 minutes for a more set cookie.

Cool on the baking sheet for 5 minutes before transferring to a wire rack. These cookies taste even better on day two when the brown butter flavor has had time to settle into the crumb.