In a small saucepan, stir together the finely diced apples, lemon juice, 3 tablespoons sugar, 1 teaspoon cinnamon, nutmeg, allspice, cornstarch, 1 teaspoon vanilla, and a pinch of salt until the apples are evenly coated.

Set the saucepan over medium heat and cook, stirring often, for 5–7 minutes, until the apples soften slightly and the juices thicken and look glossy. The apples should be tender but not mushy.

Transfer the apple mixture to a shallow bowl or plate and spread it out. Let it cool completely to room temperature so it doesn’t melt the butter in the cookie dough.

In a large mixing bowl, beat the softened butter, 3/4 cup granulated sugar, and 1/4 cup packed light brown sugar with a hand mixer or stand mixer on medium speed for 2–3 minutes, until light and fluffy. Scrape down the bowl as needed.

Beat in 1 large egg and 1 teaspoon vanilla extract until fully combined and creamy.

In a separate bowl, whisk together the flour, baking powder, 1/2 teaspoon cinnamon, and fine sea salt until well combined.

Add the dry ingredients to the butter mixture in two additions, mixing on low speed just until no streaks of flour remain. The dough should be soft but not sticky; if it is very sticky, mix in 1–2 tablespoons more flour.

Cover the bowl and chill the dough in the refrigerator for at least 30 minutes and up to 24 hours. Chilling helps the cookies keep their shape and develop a slightly flaky, rich texture.

When ready to bake, preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper or silicone baking mats.

Scoop about 1 1/2 tablespoons of chilled dough per cookie and roll into balls. Place on prepared baking sheets about 2 inches apart, then gently flatten each ball into a thick disk with your fingers or the bottom of a glass.

Use your thumb or the back of a teaspoon to press a shallow indentation in the center of each disk to form a small well for the apple filling.

Spoon about 1–2 teaspoons of cooled apple filling into each well, gently pressing the filling in so it sits snugly. Avoid mounding too high to prevent overflow while baking.

In a small bowl, whisk together 1 large egg and 1 tablespoon milk or cream. In another small bowl, stir the coarse or granulated sugar with about 1/4 teaspoon cinnamon.

Lightly brush the exposed dough edges (not the filling) with egg wash, then sprinkle the cinnamon sugar mixture over the dough rims for a golden, slightly crisp, sugary edge.

Bake one sheet at a time for 11–14 minutes, until the edges look set and just turning golden and the bottoms are lightly browned. The centers will look soft but will firm up as the cookies cool.

Let the cookies cool on the baking sheet for about 5 minutes, then transfer to a wire rack to cool completely. If glazing, wait until the cookies are fully cool.

For a bakery-style finish, whisk together powdered sugar, 1–2 tablespoons milk or cream, and 1/4 teaspoon vanilla until smooth and pourable.

Drizzle the glaze over the cooled cookies in thin zigzags. Let the glaze set for about 15 minutes before stacking or storing.