Preheat your oven to 350°F and move the oven rack to the middle position. Line a baking sheet with parchment paper to prevent sticking. Dice the half apple into 1/4-inch cubes; peel if you prefer, but I like to leave the skin on for extra texture and color.

In a medium mixing bowl, add the 3 tablespoons of room temperature unsalted butter and 1/3 cup white sugar. Cream them together with a hand mixer on medium speed until the mixture is light and fluffy. Add in the large egg yolk, 4 tablespoons of full-fat sour cream at room temperature, and 1 teaspoon of pure vanilla extract. Continue to beat until the mixture is smooth and fully combined.

Sift (or sprinkle) the dry ingredients—1 cup all-purpose flour, 1/2 teaspoon baking powder, 1/8 teaspoon baking soda, 1/8 teaspoon salt, and 1/2 teaspoon pumpkin spice blend—over the wet mixture. Using the hand mixer on low speed, mix just until there are no dry streaks. Gently fold in the diced apples from Step 1 using a spatula. I like to sift the dry ingredients using a fine mesh strainer set over the bowl for the airiest cookies.

With a cookie scoop or spoon, portion the dough into 8 equal mounds, spacing them at least 2 inches apart on the prepared baking sheet. Bake for 14 to 16 minutes, or until the bottoms are just starting to turn light golden brown. Do not overbake for best texture. Let the cookies cool on the baking sheet on a wire rack for 10 minutes, then transfer the cookies directly to the wire rack to cool completely.

While the cookies cool, make the icing. In a small saucepan over medium heat, melt 1/2 tablespoon unsalted butter with 3 tablespoons packed dark brown sugar, stirring constantly until the sugar has dissolved and the mixture is foamy (about 2–3 minutes). Carefully whisk in 4 1/2 teaspoons whole milk, bring the mixture briefly back to a boil, then remove from heat. Stir in 1/4 cup powdered sugar and 1/4 teaspoon vanilla extract using a wooden spoon until smooth. For extra depth, I sometimes add a tiny pinch of salt here.

Drizzle the brown sugar icing from Step 5 generously over the cooled cookies. Let the icing set, then enjoy your soft apple pumpkin spice cookies!