Preheat your oven to 350°F. While it heats, shred your zucchini and squeeze it thoroughly in a clean kitchen towel or cheesecloth to remove excess moisture—this is crucial for preventing soggy cookies. Sift your icing sugar into a bowl to remove any lumps, and bring your egg and cream cheese to room temperature, which helps them incorporate smoothly into the batter and frosting later.

In a large bowl, combine the squeezed zucchini, yogurt, room temperature egg, brown sugar, sugar, and 1 tablespoon vanilla. Stir until well combined and the mixture has a uniform consistency. The yogurt adds moisture and tenderness while the zucchini keeps the cookies moist without making them dense.

In a separate bowl, whisk together the flour, baking powder, salt, cinnamon, and nutmeg. Whisking instead of just stirring ensures even distribution of the leavening agents and spices, which gives you consistent texture and flavor throughout the cookies.

Pour the dry ingredient mixture from Step 3 into the wet ingredients from Step 2, and stir until just combined—don't overmix, as this develops gluten and can make the cookies tough. A few small flour streaks are fine; they'll disappear during baking.

While the oven preheats and you prepare the batter, melt the 4 tablespoons of butter in a small pan over medium heat. Watch it carefully as it foams and the milk solids turn golden brown and nutty—this takes about 5-8 minutes. I like to use a light-colored pan so I can see the color change clearly. Once golden, pour the butter into a bowl and let it cool to room temperature; you want it soft but not liquid when mixing the frosting.

Drop spoonfuls of batter onto a parchment-lined baking sheet, spacing them about 2 inches apart to allow for slight spreading. Bake for 12-15 minutes, until the edges are lightly golden but the centers still look slightly soft—they'll continue cooking on the warm pan. Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely before frosting.

In a bowl, beat together the cooled brown butter from Step 5 and the room temperature cream cheese until smooth and creamy. Add the sifted icing sugar from Step 1 gradually, then the 3/4 teaspoon vanilla and milk, beating until you reach a spreadable consistency. The cream cheese adds a subtle tang that balances the sweetness, and I find it really elevates these cookies compared to regular butter frosting.

Once the cookies are completely cool, spread or pipe the brown butter cream cheese frosting onto each cookie. Serve immediately or store in an airtight container in the refrigerator for up to 3 days.