In a large mixing bowl or stand mixer fitted with the paddle, add the softened butter and brown sugar. Beat on medium speed for 2–3 minutes, until light, fluffy, and slightly lighter in color. Pour in the maple syrup and beat again for 1 minute; the mixture may look slightly curdled, which is okay.

Add the egg and 2 teaspoons vanilla extract to the bowl. Beat on medium speed just until fully incorporated and the mixture is smooth and creamy. Do not overmix.

In a separate medium bowl, whisk together the flour, baking soda, baking powder, 3/4 teaspoon fine sea salt, cinnamon, and nutmeg (if using) until evenly combined.

With the mixer on low speed, gradually add the dry ingredients to the wet mixture in 2–3 additions. Mix just until the flour disappears and a soft, thick dough forms. If the dough seems sticky, it will firm up during chilling.

Cover the bowl with plastic wrap or a reusable lid and refrigerate the dough for at least 30–45 minutes, or up to 24 hours. For chewier cookies, chill at least 1 hour.

When ready to bake, preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper or silicone baking mats. If the dough is very firm, let it sit at room temperature for 5–10 minutes until scoopable but not sticky.

Use a medium cookie scoop (about 1 1/2 tablespoons of dough) to portion the dough onto the prepared baking sheets, spacing about 2 inches apart. For smoother tops, roll each scoop into a ball. Bake for 9–12 minutes, until the edges look set but the centers are still soft and slightly puffy. The bottoms should be light golden and the tops may have tiny cracks.

Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely before icing.

In a medium bowl, whisk together the powdered sugar and a pinch of fine sea salt. Add 3 tablespoons maple syrup and 1/2 teaspoon vanilla extract, then whisk until thick and smooth. The mixture will be very stiff at this stage.

Add the milk or heavy cream 1 tablespoon at a time, whisking after each addition, until the icing is silky and pours in a thick ribbon from the whisk. It should be thick enough to cling to the cookies but thin enough to settle into a smooth, glossy surface. If it becomes too thin, whisk in a little more powdered sugar; if too thick, add a few more drops of milk or cream.

Once the cookies are completely cool, spoon about 1–2 teaspoons of icing onto the center of each cookie. Use the back of the spoon to gently nudge the icing toward the edges; it will slowly spread and smooth out. Let the iced cookies sit at room temperature for 20–30 minutes, or until the icing is set.