Preheat your oven to 375°F and grab two baking sheets. Beat the softened butter and 1 cup sugar together until the mixture is light and fluffy—this takes about 2 minutes with an electric mixer. You'll see it transform from a thick paste into something almost cloud-like, and that's exactly what you want.
Add your 2 eggs one at a time, beating well after each addition so they fully incorporate. Then stir in the vanilla extract until combined. The mixture should be smooth and creamy at this point, with no streaks of egg white visible.
Whisk together the flour, cream of tartar, baking soda, salt, and nutmeg in a separate bowl, making sure to break up any lumps. Add these dry ingredients to your butter mixture and stir until just combined—don't overmix or you'll toughen the dough. A few flour streaks are totally fine to leave alone.
Mix the 2 teaspoons cinnamon and 1/4 cup sugar in a shallow bowl for your coating. This is your cinnamon-sugar magic, so don't skip this step because it creates that signature crunch everyone loves.
Drop rounded tablespoons of dough onto your prepared baking sheets, spacing them about 2 inches apart. Brush each dough ball lightly with the melted butter, then press it into your cinnamon-sugar mixture until it's well coated on top and sides.
Bake for 10 to 12 minutes, until the edges are just barely golden but the centers still look a tiny bit underbaked. I know this sounds risky, but trust me—this is what keeps your snickerdoodle cookies recipe soft and chewy instead of hard as rocks. Mine always stick a little to the pan and that's totally fine.
Let them cool on the baking sheet for 5 minutes so they firm up slightly, then transfer to a wire rack. They'll continue cooking from residual heat and firm up beautifully as they cool completely.