In a large mixing bowl, whisk together the flour, cream of tartar, baking soda, cinnamon, nutmeg, and salt; set aside.

In the bowl of a stand mixer fitted with a paddle attachment, beat the butter until smooth, about 1 to 2 minutes. Add both sugars and continue beating on medium speed until smooth and creamy. Scrape down the sides of the bowl, add the egg, and beat just until combined. Add the milk and vanilla and beat on medium-high speed until light and fluffy.

Dump in the flour mixture and beat on low just until combined, about 1 minute. Cover dough tightly with plastic wrap and refrigerate for at least 30-45 minutes, or until it's less sticky, and easier to handle. In a small bowl, stir together the 1/4 cup of sugar with 1 teaspoon of cinnamon.

Preheat oven to 425ºF.

In a large bowl, combine peaches, brown sugar, lemon juice, vanilla, almond extract, cornstarch, cinnamon, nutmeg, and salt. Toss to coat peaches evenly, then pour into an ungreased 9x13-inch baking dish. Bake for 10 minutes, then remove from the oven.

Remove snickerdoodle dough from the refrigerator and spoon tablespoons of the cookie dough mixture evenly over the top in a single layer. Sprinkle cinnamon sugar mixture on top. Bake in the oven for 28-30 minutes, or until topping is golden brown and cooked through. Allow cobbler to cool for 10 minutes before serving. Serve warm with vanilla ice cream.