In a large mixing bowl, whisk the flour, cream of tartar, baking soda, cinnamon, nutmeg, and salt together; 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 fully incorporated.
Dump in the flour mixture and beat on low just until combined, about 1 minute. Wrap dough tightly with plastic wrap and refrigerate for 2 hours, or until it's less sticky, and easier to handle.
Make the Cinnamon-Sugar:
In a small bowl, stir the sugar and cinnamon together.
Make the Berry Filling:
Preheat the oven to 350ºF. In a medium saucepan, heat 6 cups of the berries over medium heat until they soften and their juices release (about 3-4 minutes).
In a separate bowl, whisk sugar, cornstarch, cinnamon, and salt together. Stir dry mixture into the berries and continue stirring until mixture thickens, and coats the back of a wooden spoon. Remove saucepan from heat and stir in the remaining 1 cup of berries and lemon juice.
Transfer the filling to an 8-inch square baking dish. Use greased hands to place flattened pieces of the snickerdoodle dough evenly over the filling. Sprinkle the cinnamon-sugar evenly on top.
Tent a piece of tin foil over the top of the cobbler. Bake for 35-40 minutes, or until bubbly and golden brown on top. Allow to cool for at least 30 minutes before serving. Leftover cobbler can be stored covered in the refrigerator for up to 5 days.