In a light-colored saucepan, melt the butter over medium heat. Continue cooking, swirling often, until it foams, then turns golden and smells nutty with browned flecks settling at the bottom, about 5–7 minutes. Immediately pour into a heatproof bowl, scraping in all the browned bits, and refrigerate until firm but still soft enough to cream, about 45 minutes, stirring once or twice.

While the butter chills, pour the apple cider into the same saucepan and bring to a simmer over medium heat. Cook until reduced to about 3 tablespoons of dark, syrupy liquid, about 12–15 minutes. Set aside to cool. A little more or less than 3 tablespoons is fine.

In a medium bowl, whisk together the flour, cream of tartar, baking soda, cinnamon, and salt. Set aside.

In a large bowl using an electric mixer, beat the chilled brown butter, brown sugar, and granulated sugar on medium speed until light and fluffy, about 2–3 minutes.

Add the eggs one at a time, beating well after each addition. Beat in the vanilla and the cooled cider syrup until combined, scraping down the sides of the bowl as needed.

Gradually add the dry ingredients and mix on low speed just until incorporated. Do not overmix.

Cover the bowl and refrigerate for 30 minutes, until the dough is firm enough to scoop and roll.

Position an oven rack in the center and preheat to 350°F. Line baking sheets with parchment paper.

In a small bowl, stir together the granulated sugar, cinnamon, and apple pie spice if using.

Using a medium (1½-tablespoon) cookie scoop, portion the dough and roll into balls. Roll each ball generously in the cinnamon-sugar until fully coated. Place on the prepared baking sheets about 2 inches apart.

Bake for 10–12 minutes, until the tops are puffed and crackled and the edges are set but the centers still look soft.

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