Cream the butter and sugar. Beat the softened butter and granulated sugar with a stand or hand mixer on medium speed for 2–3 minutes until pale, fluffy, and creamy. Proper creaming whips in air that lifts the cookies, so don't shortcut it — under-creamed butter makes flat, dense cookies.
Add the egg, vanilla, and lemon zest. Beat in the egg until fully absorbed, then the vanilla extract and lemon zest. Scrape down the sides of the bowl once so everything is evenly mixed.
Work in the dry ingredients — gently. Whisk the flour, baking powder, salt, and mixed spice together in a separate bowl. Add it to the butter mixture and mix on low just until a soft dough forms, using a sturdy metal spoon at the start and switching to your hands only to bring it together — then knead for 1–2 minutes and no more, until smooth. Working the flour any longer develops gluten and turns the cookies tough instead of tender.
Chill the dough. Flatten the dough into a thick disc, wrap it tightly in plastic wrap, and refrigerate for at least 1 hour (or up to 24 hours). Cold dough rolls without sticking and holds razor-sharp edges in the oven instead of spreading into blobs.
Roll in quarters and cut your shapes. Heat the oven to 350°F and line two baking sheets with parchment paper. Divide the dough into 4 portions, rewrap three, and keep them in the fridge. Roll one portion on a lightly floured surface to 1/4-inch thickness, cut your shapes, and space them 1 inch apart on the sheets. If you want Christmas tree cookies, cut festive shapes and poke a hole near the top of each one with a skewer before baking.
Bake to pale gold — then let them set. Bake for 10–12 minutes, until the edges are just set and faintly golden while the centers still look slightly soft. Leave the cookies on the hot sheet for 5 minutes — they firm up as they cool, and that carry-over rest is what keeps the centers chewy — then move them to a wire rack to cool completely.
Glaze and decorate. Whisk the confectioners' sugar with 2 tablespoons of milk, adding more a teaspoon at a time, until you have a smooth, thin glaze that ribbons off the spoon. Brush or spoon it over the cooled cookies, shower with colored sugar and sprinkles, and leave on the rack until the icing is fully dry, about 1 hour.