Pour 1¼ cups apple cider into a small saucepan. Bring to a gentle simmer over medium heat, then reduce the heat to medium-low. Simmer uncovered for 15 to 20 minutes, stirring occasionally, until the cider has reduced to ¼ cup. Remove from the heat and let it cool completely.
In a medium bowl, whisk together the flour, cornstarch, baking soda, cinnamon, nutmeg, allspice, and salt.
In a large bowl, beat the softened butter, brown sugar, and granulated sugar with a hand mixer or stand mixer until light and creamy, about 2 minutes.
Add the egg, vanilla extract, and 3 tablespoons of cooled reduced apple cider. Beat until smooth and fully combined.
Add the dry ingredients to the wet ingredients. Mix on low speed just until no dry flour remains. Do not overmix.
Cover the dough and refrigerate for 30 minutes.
Preheat the oven to 350°F. Line two baking sheets with parchment paper.
Scoop the dough into 1½-tablespoon portions and place them 2 inches apart on the prepared baking sheets.
Bake for 9 to 11 minutes, until the edges are set and the centers no longer look wet or glossy. Do not overbake.
Let the cookies cool on the baking sheet for 5 minutes, then transfer them to a wire rack to cool completely.
For the glaze, whisk the powdered sugar, 1 tablespoon reduced apple cider, vanilla extract, and cinnamon in a small bowl until smooth. Add regular apple cider a few drops at a time only if needed for a drizzle-able consistency.
Drizzle the glaze over the cooled cookies. Let it set for 15 to 20 minutes before serving or storing.