My family went through an oatmeal phase one brutal winter when we all wanted a nice hot bowl of oatmeal for breakfast every day. Except for Buddy Jr., who hated oatmeal. Lisa tried everything, but nothing worked, till she tried Granny Smith apples, with a little help from brown sugar and cinnamon. That combination led to these cookies, which are packed with flavor and texture, and just as welcome on a cold, wet day as those bowls of oatmeal were.
Position a rack in the center of the oven and preheat the oven to 350 degrees F.
Heat a small, heavy saucepan over medium-high heat. Add 2 teaspoons of the butter, and let melt. Add the apple, 1 teaspoon of the brown sugar, the cinnamon, and a scant pinch of salt, and stir well to combine. Cook, stirring occasionally, until the apples are slightly softened, about 3 minutes.
In the bowl of a stand mixer, cream the remaining butter and brown sugar on medium-high speed until the mixture appears light and fluffy. Add the egg and vanilla and beat until incorporated.
Combine the flour, baking soda, and 1/4 teaspoon salt in a separate bowl. With the mixer on low, beat in the dry ingredients, including the oats, then fold in the apples by hand.
Line 2 baking sheets with parchment paper. Scoop out 2 tablespoons of dough at a time and arrange them at least 2 inches apart on the prepared cookie sheet. Bake, rotating the sheets top to bottom halfway through baking, until the cookies appear set, 15 to 17 minutes. Remove and let cool for a few minutes on the sheet, then transfer to a baking rack to cool completely.
To make the optional glaze: Put the confectioners' sugar, the vanilla extract, and 1 tablespoon water in a medium mixing bowl and whisk until the mixture is smooth. Drizzle the glaze over the cookies using a spoon.