Pour 1 cup apple cider into a small saucepan over medium heat. Simmer for 15 to 20 minutes, stirring occasionally, until reduced to about 1/4 cup and slightly syrupy. Cool completely for about 15 minutes.
In a medium bowl, whisk together flour, cinnamon, nutmeg, baking soda, and salt for about 30 seconds. Set aside.
Beat softened butter, brown sugar, and granulated sugar on medium speed for 2 to 3 minutes until pale, light, and fluffy. Scrape down the bowl as needed.
Add the egg and vanilla, beating until combined. Mix in 3 tablespoons of the cooled reduced cider on low speed for about 30 seconds. Reserve the remaining 1 tablespoon for the glaze.
With the mixer on lowest speed, add the flour mixture a little at a time. Mix just until no dry streaks remain and a soft dough forms. Do not overmix.
Cover the bowl and refrigerate for 30 minutes. Meanwhile, line two baking sheets with parchment paper or silicone mats.
Preheat the oven to 350°F and position a rack in the center. In a shallow bowl, stir together 1/2 cup granulated sugar and 1 teaspoon cinnamon.
Scoop the chilled dough into about 24 portions, roll into smooth balls, and coat generously in the cinnamon sugar. Place on prepared baking sheets about 2 inches apart and press each ball slightly with your palm.
Bake one sheet at a time for 10 to 12 minutes, until the edges are set and the centers look soft and slightly puffed. Cool on the baking sheet for 5 minutes, then transfer to a wire rack.
Let cookies cool completely, about 20 minutes. Whisk powdered sugar with the remaining 1 tablespoon reduced cider until smooth, thinning with milk 1 teaspoon at a time if needed. Drizzle over cookies and let set for 15 minutes.